Geometric and Higher Order Logic in terms of Abstract ...pt/ASD/geohol/geohol.pdf1 Introduction The...

Geometric and Higher Order Logic

in terms of Abstract Stone Duality

Paul Taylor

AbstractThe contravariant powerset and its generalisations ΣX to the lattices of open subsets of a

locally compact topological space and of recursively enumerable subsets of numbers satisfy

the Euclidean principle that σ and F (σ)hArr σ and F (gt)

Conversely when the adjunction Σ(minus) a Σ(minus) is monadic this equation implies that Σ

classifies some class of monos and the Frobenius law existx(φ(x) and ψ) hArr (existxφ(x)) and ψ for the

existential quantifier

In topology the lattice duals of these equations also hold and are related to the Phoa

principle in synthetic domain theory

The natural definitions of discrete and Hausdorff spaces correspond to equality and in-

equality whilst the quantifiers considered as adjoints characterise open (or as we call them

overt) and compact spaces Our treatment of overt discrete spaces and open maps is precisely

dual to that of compact Hausdorff spaces and proper maps

The category of overt discrete spaces forms a pretopos The paper concludes with a

converse of Parersquos theorem (that the contravariant powerset functor is monadic) that charac-

terises elementary toposes by means of the monadic and Euclidean properties together with

all quantifiers making no reference to subsets

Contents 6 Discrete and Hausdorff objects 211 Introduction 1 7 Overt and compact objects 272 Support classifiers 4 8 The quantifiers 313 The Euclidean principle 9 9 Unions and coproducts 354 The origins of the Euclidean principle 15 10 Open discrete equivalence relations 375 The Phoa Principle 18 11 Monadicity for elementary toposes 42

Theory and Applications of Categories 7(15) 284ndash338 December 2000

[11 April 2011 This was chronologically the first paper in the Abstract Stone Duality pro-gramme which has this evolved quite considerably in the intervening years This version there-fore contains numerous annotations that explain how the ideas developed or didnrsquot develop Thenotation has been made to conform to that adopted in later work and minor corrections havebeen made without comment However significant comments and modifications are indicated bysquare brackets]

1 Introduction

The powerset construction was the force behind set theory as Ernst Zermelo formulated it in 1908but higher order logic became the poor relation of foundational studies owing to the emphasis onthe completeness theorem in model theory In this paper the powerset plays the leading role andwe derive the first order connectives from it in a novel way The collection of all subsets is also

1

treated in the same way as the collections of open and recursively enumerable subsets in topologyand recursion theory The underlying formulation in which we do this is a category with a ldquotruthvaluesrdquo object Σ for which the adjunction Σ(minus) a Σ(minus) is monadic Robert Pare proved in 1974that any elementary topos has this property whilst the category LKLoc of locally compact localeshas it too (sectO 510 and Theorem B 3111) [B] explains how this is an abstraction of Stone duality

Gerhard Gentzenrsquos natural deduction was the first principled treatment of the logical connec-tives and Per Martin-Lof used the CurryndashHoward isomorphism to extend it in a principled way totype constructors [ML84] However the natural conclusion of such an approach is to say that theexistential quantifier is the same as the dependent sum ie that a proof of existx φ(x) must alwaysprovide a witness a particular a for which φ(a) holds

This conflicts with geometrical usage in which we may say that the Mobius band has two edgesor a complex number two square roots locally but not globally ie there exists an isomorphismbetween 2 equiv 1 + 1 and the set of edges or roots on some open subspace [Tay99 sect24] Similarlyinterprovable propositions are for Martin-Lofrsquos followers isomorphic types not equal ones andtheir account of the powerset is a bureaucratic one a structure within which to record the historiesof formation and proof of the propositionndashtypes [op cit sect95]

The way in which category theory defines the powerset is not perhaps based so firmly on alogical creed as is Martin-Lof type theory in that it describes provability rather than proof but itwas at least designed for the intuitions of geometry and symmetry This notion mdash the subobjectclassifier in an elementary topos which is readily generalised to the classifier Σ for open subsets(the Sierpinski space) and recursively enumerable ones mdash then obeys the curious equation

σ and F (σ)hArr σ and F (gt) for all σ isin Σ and F Σrarr Σ

which we call the Euclidean principle The Frobenius law which is part of the categoricalformulation of the (geometrical) existential quantifier and was so called by Bill Lawvere is anautomatic corollary of the Euclidean principle From this we develop the connectives of first ordercategorical logic in particular stable effective quotients of equivalence relations

Whilst set theory and topology have common historical roots [Hau14] the motivation for acommon treatment of the kind that we envisage is Marshall Stonersquos dictum that we should ldquoalwaystopologizerdquo mathematical objects even though they may have been introduced entirely in termsof discrete ideas [Joh82 Introduction] For example the automorphisms of the algebraic closureof Q form not an infinite discrete (Galois) group but a compact topological group Similarly thepowerset of even a discrete set is not itself a discrete set but a non-Hausdorff topological lattice(Steven Vickers has taken the same motivation in a different direction [Vic98])

The types in our logic are therefore to be spaces The topological structure is an indissolublepart of what it is to be a space it is not a set of points together with a topology any more thanchipboard (which is made of sawdust and glue) is wood

When we bring (not necessarily Martin-Lof) type theory together with categorical logic [Tay99]logical notions such as the quantifiers acquire meanings in categories other than Set In par-ticular with the internal lattice ΣX in place of the powerset of X the internal adjunctionsexistX a Σ a forallX where they exist suggest interpretations of the quantifiers We shall find thatthey obey the usual logical rules but in the topological setting they also say that the space X isrespectively overt (a word that we propose to replace one of the meanings of open) or compact

It is well known that the recursively enumerable subsets of N almost form a topology sincewe may form finite intersections and certain infinitary unions of ldquoopenrdquo subsets However theunification of topology with recursion theory ie making precise Dana Scottrsquos thesis that continuityapproximates the notion of computability involves a revolutionary change because the classical

1The letters denote the other papers in the ASD programme

2

axiomatisation of (the frame of open subsets of) a topological space demands that we may formarbitrary unions

The usual procedure for turning a base (something like this ldquotopologyrdquo on N that partiallysatisfies the axioms for a topological space) into a topology would make every subset of N openlosing all recursive information In particular in topology an open subspace is glued to its closedcomplement by means of a comma square construction that is due to Michael Artin but thisdoesnrsquot work for recursively enumerable subsets and their complements [D Section 4]

Although they are in many respects more constructive the modern re-axiomatisations of topol-ogy in terms of open subsets mdash the theory of frames or locales [Joh82] that came out of topostheory [Joh77] and Giovanni Sambinrsquos formal topology [Sam87] motivated by Martin-Lof typetheory mdash have exactly the same fault as Bourbakirsquos [Bou66]

The finite meets and joins in the theory of frames present no problem so what we need is a newway of handling the ldquopurely infinitaryrdquo directed joins Here we use an idea to which Scottrsquos namehas become firmly attached (though it goes back to the RicendashShapiro and MyhillndashShepherdsontheorems of 1955) that directed joins define a topology However we turn this idea on its head bytreating frames not as infinitary algebras over Set but as finitary ones over Sp (ie as topologicallattices cf topological groups) we can use topology in place of the troublesome directed joinswhenever they are genuinely needed (Nevertheless substantial re-working of general topology isneeded to eliminate the use of interiors Heyting implication direct images nuclei and injectivity)

We do this by postulating that for the category C of spaces the adjunction

Cop

C

Σ(minus)

6a Σ(minus)

be monadic ie Cop is equivalent to the category of EilenbergndashMoore algebras and homomorphismsover C In practice this is used in the form of Jon Beckrsquos theorem about U -split coequalisers Theway in which this expresses both Stone duality and the axiom of comprehension is explored inSections B 1 and B 8 The concrete topological model of this situation is the category of locallycompact spaces (or locales) and continuous maps

But this category does not have all equalisers pullbacks and coequalisers The impact of thison logic is that we must also reconsider the notions of equality and inequality which we defineby saying that the diagonal subspace is respectively open or closed ie that the space is discreteor Hausdorff

[It will emerge later in the ASD programme that we cannot after all do without equalisersand pullbacks especially when we consider recursion in [E Section 2]]

Computational considerations also urge such a point of view Two data structures may repre-sent the same thing in some ethereal mathematical sense for example in that they encode functionsthat produce the same result for every possible input value However as we are unable to testthem against every input such an equality may be outside our mortal grasp A similar argumentapplies to inequality or distinguishing between the two data structures in particular real numbersEquality and inequality therefore are additional structure that a space may or may not possess

Now that equality is no longer to be taken for granted as has traditionally been done in puremathematics there are repercussions for category theory Specifically Peter Freydrsquos unificationof products kernels and projective limits into the single notion of limit in a category [Fre66b]breaks down because the non-discrete types of diagram depend on equality This entails a root-and-branch revision of categorical logic which has traditionally relied very heavily on the universalavailability of pullbacks In fact most of this work has already been done in categorical type theory[Tay99 Chapters VIII and IX]

3

The duality between equality and inequality (which also relates conjunction to disjunctionand universal to existential quantification) is the characteristic feature of classical logic for allits other merits intuitionistic logic loses it However we find that it reappears in topology andrecursion for everything that we have to say in this paper about conjunction equality existentialquantification open subsets and overt spaces we find exactly analogous results for disjunctioninequality universal quantification closed sets and compact spaces

This symmetry is a constructive theorem (for LKLoc) as a result of Scott continuity theEuclidean principle implies its lattice dual Together with monotonicity (the finitary part of Scottcontinuity) the two Euclidean principles amount to the Phoa2 principle that has arisen insynthetic domain theory [Pho90a Hyl91 Tay91] These are also theorems in the free modelof the other axioms In view of their novelty and unusual form connections to the Markovprinciple and several other things have deliberately been left as loose ends

Whilst Scott continuity is obviously an important motivating principle the Phoa principle alonehas been enough to develop quite a lot of general topology keeping the openndashclosed symmetry aprecise one so far

In particular we have a completely symmetrical treatment of open and proper maps includ-ing the dual Frobenius law identified by Japie Vermeulen Currently it only deals with inclusionsand product projections but the analogue and lattice dual of Andre Joyal and Myles Tierneyrsquosldquolinear algebrardquo for locales [JT84] will be developed in future work

The topics in general topology that we discuss in the body of the paper converge on a treatmentof overt discrete spaces (classically these are sets with the discrete topology) showing that theyform a pretopos That is they admit cartesian products disjoint unions quotients of equivalencerelations and relational algebra [Assuming Scott continuity they also admit free monoids and soform an arithmetic universe [E]]

The whole of the paper therefore concerns the logic of the category of sets even though muchof it is written in topological language what we say about the subcategory of overt discrete spacesis immediately applicable to the whole category when this is Set or an elementary topos In thissense we have a new account of some of the early work on elementary toposes in particular thatthey satisfy Jean Giraudrsquos axioms ie that any topos is also what we now call a pretopos

The distinction between topology and set theory turns out therefore to be measured by thestrength of the quantifiers that they admit The paper concludes with a new characterisation ofelementary toposes that is based like Parersquos theorem on monadicity of the contravariant powersetbut which makes no reference whatever to subsets

2 Support classifiers

We begin with the way in which powersets are defined in topos theory ie using the subobjectclassifier (Section 11) and exponentials but expressed in a slightly more flexible way We mayarrive at the same definitions from type-theoretic considerations [Tay99 sect95] But whereas theuniqueness of the characteristic map φ is a moot point in that discipline it is essential to thispaper

The subobject classifier was originally defined by Bill Lawvere in 1969 and the basic theory ofelementary toposes was developed in collaboration with Myles Tierney during the following year[Law71 Law00] Giuseppe Rosolini generalised the definition to classes of supports [Ros86] anddeveloped a theory of partial maps but the Frobenius law (Propositions 311 82 and 1013) is

2This is an aspirated p not an f Wes Phoa told me that his name should be pronounced like the French wordpoire ie pwahr though maybe this is only helpful to the southern English and his fellow Australians as there isno final r

4

also required for relational algebra

Definition 21 A class M of morphisms (written rarr) of any category C such that(a) all isomorphisms are in M

(b) all M-maps are mono so i isinM must satisfy i a = i brArr a = b

(c) if i X rarr Y and j Y rarr Z are in M then so is j i and

V - U

Y

flowasti

cap f - X

i

cap

(d) if i U rarr X is in M and f Y rarr X is any map in C then the pullback flowasti V rarr Y existsin C and belongs to M

is called a class of supports or a dominion

Definition 22 An M-map gt 1rarr Σ is called a support classifier or a dominance (for M)if for every M-map i U rarr X there is a unique characteristic map φ X rarr Σ making thesquare a pullback

U - 1

X

i

cap

φ- Σ

gt

Set-theoretically U is obtained from φ as x X | φ[x] by the axiom of comprehension (separa-tion or subset-selection) though we shall find the abbreviation [φ] rarr X convenient here Becauseof the topological intuition we call i U rarr X (the inclusion of) an open subset (We shall write- for closed subsets and for Σ-split ones)

Notice that this pullback is also an equaliser

U subi - X

gt -

φ- Σ

The relationship between the axiom of comprehension and the monadic ideas of this paper isexplored in Section B 8 which also sets out the λ-calculus that we need in a more explicitlysymbolic fashion

Remark 23 This classification property deals only with a single subobject of (or predicate on)X but in practice we need to consider Γ-indexed families of subobjects or predicates containingparameters ~z whose types form the context Γ These may equivalently be seen as binary relationsΓ X or as subobjects of ΓtimesX which are classified by maps ΓtimesX rarr Σ We would like theseto be given by maps from Γ ie by generalised elements of the internal object of maps X rarr ΣHence we want to use the exponential ΣX In Set this is the same as the powerset P(X) Tosummarise there is a correspondence amongst

i~z U~z rarr X φ~z[x] φ ΓtimesX rarr Σ and φ Γrarr ΣX equiv P(X)

where φ(~z ) is the subset U~z for (~z ) isin Γ

5

Remark 24 We do not intend C to be cartesian closed because this does not follow from ouraxioms and we want to use the category of locally compact spaces as an example Recall that tosay that the exponential Y X exists in a category C (as a property of these objects individually)means that the functor C(minus timesXY ) Cop rarr Set is representable ie it is naturally isomorphicto C(minus Z) for some object Z which we rename Z equiv Y X For this to be meaningful all binaryproducts ΓtimesX must first exist in C

Notice that we ask for exponentials of the form Y X for all X and fixed Y whereas the wordexponentiability refers to a property of a particular object X for arbitrary Y Peter Freyd [FS90]has used the word baseable for the property that we require of Σ but recognising the meaningof that word out of context depends on already knowing its association with exponents whereasexponentiating suggests its own meaning more readily (The word exponent seems to have comefrom the French exposant [Bar88])

Remark 25 We shall make use of the λ-calculus to define morphisms to and from exponentialssuch as these However since we are only assuming the existence of ΣX and not cartesianclosure the body of any λ-expression that we use must be of type Σ or some other provablyexponentiating object such as a retract of ΣY The range ie the type of the bound variable isarbitrary (cf Definition 77)

[The restricted λ-calculus that we require ie with just ΣY instead of general exponentials issketched in Section A 2]

We leave it to the reader making use of some account of λ-calculus and cartesian closedcategories such as [Tay99 sect47] to rewrite juxtapositions like φ(fy) categorically in terms ofevaluation (ev ΣX times X rarr Σ) and λ-abstractions as adjoint transpositions When we writex isin X φ isin ΣX etc we mean generalised elements ie C-morphisms x Γrarr X and φ Γrarr ΣX or expressions of type X or ΣX involving parameters whose types form an unspecified contextΓ to make Γ explicit in our categorical expressions often involves forming products of variousobjects with Γ On the other hand when we write f X rarr Y it is sufficient for our purposes toregard this as a particular C-morphism ie a global element of its hom-set though in most casesparametrisation is possible by reading f as a morphism Γ times X rarr Y At first we need to makethe parametrising object explicit (calling it X) as we are considering a new logical principle butfrom Section 6 it will slip into the background

Symbolically our λ-calculus is peculiar only in that there is a restriction on the applicabilityof the (rarr)-formation rule such a calculus has been set out by Henk Barendregt [Bar92 sect52]although this is vastly more complicated than we actually need here We shall instead adopt amuch simpler notational convention lower case Greek letters capital italics Σ(minus) equiv (minus)lowast and thelogical connectives and quantifiers denote terms of exponentiating type (predicates) whilst lowercase italics denote terms of non-exponentiating type (individuals and functions) The capitalitalics could be thought of as ldquogeneralised quantifiersrdquo

The objects unlike the morphisms are never parametric in this paper (except in Proposi-tion 54)

[We also refer to an equation between λ-terms as a statement]

Remark 26 As exponentials are defined by a universal property the assignment X 7rarr ΣX

extends to a contravariant endofunctor Σ(minus) C rarr Cop It takes f Y rarr X to

Σf ΣX rarr ΣY by Σf (φ) equiv φ f equiv f φ equiv λy φ(fy)

6

The effect of Σf is to form the pullback or inverse image along f

V - U - 1

Y

cap f - X

cap

φ - Σ

gt

Remark 27 Returning to the powerset we write SubM(X) for the collection of isomorphismclasses of M-maps into X isin obC Then Definition 22 says that the contravariant functor SubM Cop rarr Set is representable ie that it is naturally isomorphic to C(minusΣ) for some object Σ Asa special case of the conditions on M the pullback (intersection) of any two M-maps into Xexists and the composite across the pullback square is in M so SubM(X) is a semilattice Infact SubM(minus) is a presheaf of semilattices on the category C so C(minusΣ) is an internal semilatticein the topos of presheaves [Fre66a] Since the Yoneda embedding is full and faithful and preservesproducts Σ was already an internal semilattice in C

It is well known that this argument makes unnecessary use of the category of sets which itis the whole point of this paper to avoid (there are also size conditions on C) although we maysuppose instead that C is an internal category in some pretopos However it is not difficult todisentangle this result from the sheaf theory and prove it directly instead (The details of thisproof would be a useful exercise in diagram-chasing for new students of category theory)

Proposition 28 Any dominance Σ carries a and-semilattice structure and pullback along gt rarr Σinduces an external semilattice isomorphism [ ] C(XΣ)rarr SubM(X)

Σtimes Σtimes Σidtimes and- Σtimes Σ Σtimes Σ

tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-

Moreover each ΣX is an internal and-semilattice and each Σf is a semilattice homomorphism (Weshall consider the existence and preservation of joins in Section 9)

1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7

Proof The fifth diagram defines and as sequential and in terms of composition inM manipulationof the pullbacks in the sixth establishes the relationship with intersection By construing the monos〈gtgt〉 rarr Σ times Σ and 〈gtgtgt〉 rarr Σ times Σ times Σ as pullbacks in various ways the semilatticelaws follow from the uniqueness of their characteristic maps These laws are expressed by thefour commutative diagrams above involving products ie Σ is an internal semilattice MoreoverΣX is also an internal semilattice because the functor (minus)

X(that is defined for powers of Σ and

all maps between them) preserves products and these commutative diagrams Similarly Σf is a

homomorphism because (minus)f

is a natural transformation

Remark 29 In the next section we give a new characterisation of support classifiers as semilatticessatisfying a further equation (the Euclidean principle) rather than by means of a class of supportsThis characterisation depends on another hypothesis that the adjunction

Σ(minus) a Σ(minus)

(which is defined for any exponentiating object Σ) be monadic The way in which this hypothesisis an abstract form of Stone duality is explored in Examples B 1 In many cases the categoryC that first comes to mind does not have the monadic property but [B] constructs the monadiccompletion C of any category C with an exponentiating object Σ In fact C is the opposite of thecategory Alg of EilenbergndashMoore algebras for the monad

Ultimately our interest is in developing some mathematics according to a new system of axiomsie in the free model (Remark 38 Theorem 42) but first we introduce the concrete situations onwhich the intuitions are based They and Example 45 also provide examples and counterexamplesto gauge the force of the Euclidean and monadic principles Even when C is not monadic itsproperties are usually close enough to our requirements to throw light on the concepts withoutshifting attention to the more complicated C

[In fact most of the results of this paper can be developed in the similar but alternativeframework of equideductive logic without the monadic principle [DD]]

Examples 210(a) Let C be Set or any elementary topos and Σ equiv Ω its subobject classifier which is an internal

Heyting algebra classically it is the two-element set So ΣX equiv P(X) is the powerset of Xand M consists of all monos 1ndash1 functions or (up to isomorphism) subset inclusions Allobjects are compact overt and discrete in the sense of Sections 6ndash8 Section 11 proves Parersquostheorem that the adjunction Σ(minus) a Σ(minus) is monadic

(b) Let C be any topos and Σ equiv Ωj be defined by some LawverendashTierney topology j so Mconsists of the j-closed monos [LR75] [Joh77 Chapter 3] [BW85 sect61] The full subcategoryEj of j-sheaves is reflective in E and the sheaves are the replete objects [BR98] Restricted toEj the adjunction is monadic and Ej is the monadic completion of E

Examples 211(a) Let Σ be the Sierpinski space which classically has one open and one closed point the

open point classifies the class M of open inclusions Then the lattice of open subsets of anytopological space X itself equipped with the Scott topology has the universal property of theexponential ΣX so long as we restrict attention to the category C of locally compact soberspaces (LKSp) or locales (LKLoc) and continuous maps [Joh82 sectVII 47ff] In this case thetopology is a continuous lattice [GHK+80] A function between such lattices is a morphism inthe category ie it is continuous with respect to this (Scott) topology iff it preserves directedjoins [Joh82 Proposition II 110]

8

With excluded middle all objects are overt but (unlike Set) LKSp and LKLoc are notcartesian closed complete or cocomplete For LKLoc the adjunction Σ(minus) a Σ(minus) is monadic(sectO 510 and Theorem B 311) as it is for LKSp assuming the axiom of choice

(b) Let C be the category Cont of continuous lattices and functions that preserve directed joinsand Σ be the Sierpinski space then ΣX is the lattice of Scott-open subsets of X In thiscase Σ(minus) is not monadic but LKLoc is its monadic completion (C) As Cont is cartesianclosed but LKLoc is not this shows that the construction of C from C in Section B 4 doesnot preserve cartesian closure For trivial reasons all objects of Cont are overt and compact

(c) In these examples we may instead takeM to be the class of inclusions of closed subsets exceptthat now the closed point perp of the Sierpinski space performs the role of gt in Definition 22(Corollary 55ff)

It is by regarding (locally compact) frames as finitary algebras over the category of spaces (inwhich directed joins are ldquopart of the wallpaperrdquo) rather than as infinitary ones over the categoryof sets that we achieve complete openndashclosed duality for the ideas discussed in this paper

Examples 212(a) Let C be Pos and Σ equiv Υ equiv Ω be the subobject classifier (regarded classically as the posetperp 6 gt) so ΥX is the lattice of upper sets cf the notation for the Alexandroff topologyin [Joh82 sectII 18] The category C is cartesian closed and has equalisers and coequalisersAlthough C rarr C is (classically) full and faithful the real unit interval [0 1] is an algebra thatis not the lattice of upper sets of any poset [FW90 Example 9] see also Example B 312

(b) The algebras for the monad are completely distributive lattices but the intuitionistic definitionof the latter is itself a research issue [FW90] so we assume excluded middle in our discussionof this example Nevertheless Francisco Marmolejo Robert Rosebrugh and Richard Woodhave shown that the opposite of the category of constructively completely distributive latticesis monadic [MRW02] Classically this category is equivalent to the category of continuousdcpos and essential Scott-continuous maps ie those for which the inverse image preservesarbitrary meets as well as joins [Joh82 sectVII2] All objects are overt and compact whilst Setis embedded as the full subcategory of discrete objects (Example 614)

In any given category there may be many classesM of supports each class possibly being classifiedby some object ΣM

Remark 213 Many of the ideas in this paper evolved from synthetic domain theory a modelof which is a topos (with a classifier Ω for all monos) that also has a classifier Σ for recursivelyenumerable subsets [Ros86 Pho90a Pho90b Hyl91 Tay91 FR97 BR98] In this case Σ is asubsemilattice of Ω Such models exist wherein the full subcategory of replete objects satisfies themonadicity property discussed in this paper for Σ [Theorem I 1510] in addition to that for thewhole category for Ω [RT98]

3 The Euclidean principle

We now give a new characterisation of dominances in terms of the object Σ (and its powers) alonewithout any reference to subsets These are supplied by the monadic assumption which thereforesomehow plays the role of the axiom of comprehension [This role is formalised in Section B 8]

Proposition 31 In any dominance Σ the Euclidean principle

φ(x) and F(x φ(x)

)lArrrArr φ(x) and F (xgt)

9

holds for all φ X rarr Σ and F X times Σrarr Σ

- [φ] cap [Fgt] sub - [Fgt] - 1

[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6

Proof The composites [φ] rarr X rArr X times Σ rarr Σ are equal by the construction of [φ] rarr X sotheir pullbacks [φ]cap [Fgt] and [φ]cap [Fφ] along gt are isomorphic ie equal as subobjects of [φ] andso of X Since [ ] C(XΣ)rarr SubM(X) preserves cap these subobjects are [φ and Fgt] and [φ and Fφ]respectively But then by the uniqueness of the characteristic map the equation holds

Another construction that we can do with a dominance will turn out to be the existentialquantifier [For this reason the letter E was used for it in the published version of this sectionbut this has been changed to I here because of the convention that was adopted later in the ASDprogramme of using E for a nucleus]

Lemma 32 Using Definition 22 let i U rarr X be the mono classified by φ X rarr Σ Then theidempotent (minus) and φ on ΣX splits into a homomorphism Σi and another map I ΣU ΣX withΣi middot I = idΣU and I middot Σi = (minus) and φ ΣX rarr ΣX

W - V - 1 ============= 1

U times ΣX

cap

idtimes Σi- U times ΣU

cap

evU - Σ

gt

X times ΣX

itimes id

cap

idtimes Σi- X times ΣU

itimes id

cap

I - Σ

gt

id

-

Proof Let V and W be the middle and left pullbacks in the top row and let I be the classifyingmap for V rarr X times ΣU so the big square is also a pullback

Now consider how V rarr U times ΣU is classified by maps targeted at the lower right corner Oneclassifier is id evU However whilst V was defined as the pullback of I against gt it is also thepullback of the composite I (i times id) against gt where the relevant triangle commutes becauseitimes id is mono Hence I (itimes id) also classifies V By uniqueness the trapezium commutes andthe exponential transposes are Σi middot I = id

The composite along the bottom is the transpose of I middotΣi The lower left square is a pullbackso the whole diagram is a pullback and W is (x ψ) isin X times ΣX | I(Σiψ)(x) However using thesmaller pullback rectangle it is (x ψ) | x isin U and ψ(x) where (x isin U) means φ(x) Again byuniqueness of (pullbacks and) the classifier these are equal so I middot Σi = (minus) and φ

10

The significance of the Euclidean principle and the map I is that they provide an exampleof the condition in the theorem of Jon Beck that characterises up to equivalence the adjunctionbetween the free algebra and underlying set functors for the category of algebras of a monad[Mac71 sectIV 7] [BW85 sect33] [Tay99 sect75] Although the public objective of the theory of monadsis as a way of handling infinitary algebra this condition will turn out to be more important inAbstract Stone Duality than the notion of algebra

Lemma 33 Let Σ be an exponentiating semilattice that satisfies the Euclidean principle (ie theconclusion of Proposition 31) Then the parallel pair (u v) in the middle of that diagram

U equiv [φ] subi

- Xsub

u x 7rarr 〈xgt〉 -

v x 7rarr 〈x φ(x)〉- X times Σ

ΣU

Σigt-I- ΣX

J ψ 7rarr λxσ σ and ψx - ΣXtimesΣ

is Σ-split in the sense that there is a map J as shown such that

(Jψ)(ux)hArr ψ(x) and J(F u)(vx)hArr J(F v)(vx)

for all x isin X ψ isin ΣX and F isin ΣXtimesΣ (We just mark u with a hook as a reminder that theseequations are not symmetrical in u and v)

Proof For ψ isin ΣX we have (Jψ)(ux)hArr gtandψ(x)hArr ψ(x) and (Jψ)(vx)hArr φ(x)andψ(x) HenceJ(F u)(vx) hArr φ(x) and F (xgt) and J(F v)(vx) hArr φ(x) and F

(x φ(x)

) which are equal by the

Euclidean principle

Remark 34 The map I defined in Lemma 32 is the fifth one needed (together with Σu and Σv)to make the lower diagram in Lemma 33 a split coequaliser

idΣU = Σi middot I I middot Σi = (minus) and φ = Σv middot J

[The term E equiv I middot Σi equiv λψ ψ and φ is a nucleus (Definition B 43 and sectO 83)

E(λx F(λψ ψx)

)= λx φx and F(λψ ψx)

= λx φx and F(λψ ψx and φx)

= E(λx F(λψ ψx)

)using the Euclidean principle with σ equiv φx Then

U equiv X | E and x isin U a` φxhArr gt a` forallψ EψxhArr ψx

in the notation of Section B 8]

Remark 35 Since Σ is a semilattice it carries an internal order relation Some morphisms aremonotone with respect to this order but others may not be The order also extends pointwiseto an order on morphisms between (retracts of) powers of Σ We shall discuss monotonicity andsome other ways of defining order relations in Section 5

The order also allows us to talk of such morphisms as being adjoint L a R

11

When the definition of adjunction is formulated as the solution of a universal property theadjoint is automatically a functor However the definition that is most useful to us is the one thatuses the internal order relation and the structure of the category directly namely

id 6 R middot L L middotR 6 id

which does not itself force L and R to be monotone So we say so even though we have nointention of considering non-monotone adjunctions This point is significant in Lemma 37 inparticular To repeat Proposition 28

Lemma 36 For any object Z the functor (minus)Z

(defined on the full subcategory of retracts ofpowers of an exponentiating semilattice Σ) preserves the semilattice structure and hence the order(ie it is monotone or order-enriched) and adjointness (LZ a RZ)

Lemma 37 Let P A rarr S be a homomorphism between internal semilattices and I S rarr Aanother morphism such that idS = P middot I Then the following are equivalent(a) I middot P = (minus) and φ for some φ 1rarr A

(b) I is monotone and satisfies the Frobenius law

I(θ) and ψ = I(θ and Pψ) for all θ isin S ψ isin A

(c) [P and I define an isomorphism A sim= S darr φ equiv ψ S | ψ 6 φ]In this case I preserves binary and so it is monotone and I a P Since the [order relation 6derived from a semilattice structure is] antisymmetric [ie φ 6 ψ ψ 6 φ ` φ = ψ] adjoints areunique so each of P I and φ uniquely determines the other two (See Definition 104 for when Ipreserves gt)

Proof [arArrb] I(θ) and ψ = I middot P middot I(θ) and ψ = (Iθ and φ) and ψ whilst

I(θ and Pψ) = I(P middot I(θ) and Pψ

)= I middot P

(I(θ) and ψ

)= (Iθ and ψ) and φ

since P preserves and[alArrb] I middot P (ψ) = I(gt and Pψ) = I(gt) and ψ so φ = I(gt)In particular I(θ1)and I(θ2) = I

(θ1 and P middot I(θ2)

)= I(θ1 and θ2) Finally I a P because idS 6 P middot I

and I middot P 6 idA

Now we apply the monadic property to logic for the first time

Remark 38 Instead of taking the class M of monos as fundamental from now on we shallassume that(a) the category C has finite products and splittings of idempotents

(b) Σ is an exponentiating object

(c) the adjunction Σ(minus) a Σ(minus) is monadic

(d) (Σgtand) is an internal semilattice and

(e) it satisfies the Euclidean principle

Lemma 39 With the assumptions in Remark 38 let φ X rarr Σ in C Then(a) the pullback i U rarr X of gt 1rarr Σ against φ exists in C(b) there is a map I ΣU rarr ΣX such that idΣU = Σi middot I and I middot Σi = (minus) and φ ΣX rarr ΣX

(c) the classifying map φ X rarr Σ is uniquely determined by i

(d) if j V rarr U is also open then so is the composite V rarr U rarr X

12

So we have a class M of monos with an exponentiating classifier Σ as in Section 2

V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-

Proof The pullback is given by an equaliser Lemma 33 shows that this pair is Σ-split using theEuclidean principle so the equaliser exists by Beckrsquos theorem and the contravariant functor Σ(minus)

takes it to the (split) coequaliser Thus the idempotent (minus)andφ on ΣX splits into a homomorphismΣi ΣX ΣU and a map I satisfying the equations above By Lemma 37 I a Σi and thecharacteristic map φ equiv I(gt) is unique Finally if V is classified by ψ U rarr Σ then the compositei j is classified by I(ψ) [See sectO 84 for a more detailed proof of (d)]

Theorem 310 Let (CΣ) satisfy the first four axioms in Remark 38 Then(a) gt 1 rarr Σ is a dominance (where M is the class of pullbacks of this map and and is given by

Proposition 28) iff

(b) (Σgtand) satisfies the Euclidean principleIn this case a mono i is classified (open) iff there is some map I that satisfies id = Σi middot I and theFrobenius law

Proof [More explanation of how the Euclidean principle entails uniqueness of classifiers isneeded This is trivial in the case of gt since the square

V - 1

U

sim=

cap

ψ - Σ

gt

is only a pullback (indeed commutes) when ψ = gt More generally if φ ψ X rArr Σ have the samepullback U then so does φ and ψ (cf Proposition 28) so without loss of generality ψ 6 φ Usingthe foregoing argument this situation transfers from X to U The role of the Euclidean principleor more precisely the Frobenius law is to make ΣU sim= ΣX darr φ (Lemma 37) The external form ofthis relationship is the top row of the diagram

C(UΣ)-

sim=(minus) middot i

C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback

========= V isin Sub(X) | V sub U

pullback

-

(minus) cap USub(X)

pullback

13

in which the bottom row says the same thing for (open) subspaces using the fact that openinclusions compose (Lemma 39(d)) We also know that the diagram commutes in the obviousways Hence if ψ 6 φ X rArr Σ both classify U then ψ factors through U rarr Σ and as such itclassifies the same open subspace of U as gt does namely U itself so ψ = gt U rArr Σ SinceC(UΣ) sim= C(XΣ) darr φ we have ψ = φ X rArr Σ]

This result partly answers a criticism of [Tay91] that it did not ask for a dominance since theEuclidean principle is a part of the Phoa principle (Proposition 57) although monadicity was notconsidered there

It does not seem to be possible in general to deduce id = Σi middot I from the simpler conditionthat i be mono (with I a Σi) but see Corollary 103 when the objects are overt and discrete

The next result justifies the name existi for I [Tay99 sect93] although it only allows quantificationalong an open inclusion we consider the more usual quantifier ranging over a type in Sections 7ndash8

Proposition 311 For i U rarr X open Σi and existi satisfy(a) the Frobenius law

existi(θ) and ψ = existi(θ and Σiψ)

for any θ isin ΣU and ψ isin ΣX and

(b) the BeckndashChevalley condition

Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi

for any map f Y rarr X in C ie that if the square consisting of g i f and j is a pullbackthen that on the right commutes

Proof [a] Lemma 37 [b] Put ω equiv existi(θ) so θ = Σi(ω) and ω = existiΣiω = ω and φ ThenexistjΣgθ = existjΣgΣiω = existjΣj(Σfω) = Σfωand (φ f) = Σf (ωandφ) = Σfω = Σfexistiθ since φ f classifiesj V rarr Y

Remark 312 Consider the pullback (intersection) U cap V sub X of two open subsets

X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---

In the diagram on the right the monos are existential quantifiers and the epis are inverse imageswhich are adjoint and split The BeckndashChevalley condition for this pullback says that the squaresfrom ΣU to ΣV and vice versa commute These equations make the square of existential quantifiers

14

an absolute pullback ie it remains a pullback when any functor is applied to it and similarlythe square of inverse image maps an absolute pushout [Tay99 Exercise 53]

Remark 313 The Euclidean equation can be resolved into inequalities

σ and F (ψ) =rArr F (σ and ψ) σ and F (σ and ψ) =rArr F (ψ)

for all σ isin Σ F ΣY rarr ΣX and ψ isin ΣY The first says that F is strong and the second is asimilar property for order-reversing functions At the categorical level the contravariant functorΣ(minus) also has such a co-strength X times ΣXtimesY rarr ΣY given by (x ω) 7rarr λy ω(x y)

4 The origins of the Euclidean principle

This characterisation of the powerset offers us a radically new attitude to the foundations of settheory the notion of subset is a phenomenon in the macroscopic world that is a consequence of apurely algebraic (ldquomicroscopicrdquo) principle and is made manifest to us via the monadic assumptionTaking this point of view where does the Euclidean principle itself come from

Remark 41 The name was chosen to be provocative The reason for it is not the geometry butthe number theory in Euclidrsquos Elements Book VII with f n isin Σ equiv N gt equiv 0 and and equiv hcf wehave a single step of the Euclidean algorithm

hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n

)for any polynomial F Nrarr N since F (n)minusF (0) = ntimesG(n)

for some polynomial G I do not know what connection if any this means that there is betweenhigher order logic and number theory the generalisation is not direct because ring homomorphismsneed not preserve hcf whereas lattice homomorphisms do preserve meets

The way in which the Euclidean principle (and the Phoa principle in the next section) expressF (σ) in terms of F (perp) and F (gt) is like a polynomial or power series (Taylor) expansion exceptthat it only has terms of degree 0 and 1 in σ and being idempotent These principles may perhapshave a further generalisation of which the KockndashLawvere axiom in synthetic differential geometry[Koc81] would be another example Of course if and is replaced by a non-idempotent multiplicationin ΣX then the connection to the powerset is lost

The Euclidean algorithm may be stated for any commutative ring but it is by no means trueof them all But it is a theorem for the free (polynomial) ring in one variable and we have similarresults in logic where ΣΣ is the free algebra (for the monad) on one generator [See also thediscussion in sectO 77]

Post-Publication Note The following is only valid for terms whose free variables are of typeN or ΣN In particular F cannot be a free variable

Theorem 42 The Euclidean principle holds in the free model of the other axioms in Remark 38

Proof There is a standard way of expressing the free category with certain structure (includingfinite products) as a λ-calculus (with cut and weakening) in which the objects of the categoryare contexts rather than types [Tay99 Chapter IV] In our case C has an exponentiating internalsemilattice and the calculus is the simply typed λ-calculus restricted as in Remark 25 togetherwith the semilattice equations The calculus gives the free category because the interpretation or

15

denotation functor JminusK C rarr D is the unique structure-preserving functor to any other categoryof the same kind

The free category C in which the adjunction Σ(minus) a Σ(minus) is monadic will be constructed (andgiven its own λ-calculus) in Section B 4 from which (for the present purposes) we only need toknow that C(minusΣΣ) sim= C(minusΣΣ)

So we just have to show that φandF (φandψ)hArr φandF (ψ) for all φ ψ isin Σ by structural inductionon F isin ΣΣ in the appropriate λ-calculus considering the cases where F (σ) is(a) gt or another variable τ trivial

(b) σ itself by associativity and idempotence of and

(c) Gσ andHσ by associativity commutativity and the induction hypothesis

(d) λx G(σ x) using φ and λx G(x σ) = λx(φ andG(x σ)

)and the induction hypothesis

(e) G(H(σ) σ

) put ψprime equiv Hψ and ψprimeprime equiv H(φ and ψ) so

φ and ψprime equiv φ andHψ hArr φ andH(φ and ψ) equiv φ and ψprimeprime

by the induction hypothesis for H and

φ andG(Hψψ) equiv φ andG(ψprime ψ)

hArr φ andG(φ and ψprime φ and ψ)

hArr φ andG(φ and ψprimeprime φ and ψ)

hArr φ andG(ψprimeprime φ and ψ)

equiv φ andG(H(φ and ψ) φ and ψ

)by the induction hypothesis for (each argument of) G

Remark 43 The result remains true for further logical structure because of generalised distribu-tivity laws(a) or and perp assuming the (ordinary) distributive law

(b) rArr by an easy exercise in natural deduction

(c) exist assuming the Frobenius law

(d) forall since it commutes with φ and (minus)

(e) N since φ and rec(nG(n) λmτ H(nm τ)

)hArr rec

(n φ andG(n) λmτ φ andH(nm τ)

)

(f) =N and 6=N since these are independent of the logical variable

(g) Scott continuity as this is just an extra equation (Remark 711) on the free structure butCorollary 55 proves the dual Euclidean principle more directly in this case

Remark 44 Todd Wilson identified similar equations to the Euclidean principle in his studyof the universal algebra of frames [Wil94 Chapter 3] in particular his Proposition 96(b) andRemark 924 although the fact that frames are also Heyting algebras is essential to his treatment

[Some other similar results are also known about maps F Ωrarr Ω(a) Denis Higgs quoted in [Joh77 Exercise I33] showed that if F is mono (but need not preserve

order) then F 2 = idΩ

(b) Jean Benabou quoted in [Woo04 sect41] showed that if F 6 id then Fσ hArr σ and FgtSee also the discussion in sectO 77 Besides these Anna Bucalo Carsten Furhmann and AlexSimpson [BFS03] investigated an equation similar to the Euclidean principle that is obeyed by thelifting monad (minus)perp]

Example 45 So-called stable domains (not to be confused with the stability of propertiesunder pullback that we shall discuss later) provide an example of a dominance that is instructive

16

because many of the properties of Set and LKSp fail Although the link and specialisation orders(Definition 59) coincide with the concrete one they are sparser than the semilattice order Allmaps Σrarr Σ are monotone with respect to the semilattice order but not all those ΣΣ rarr Σ Theobject Σ is not an internal lattice so there is no Phoa or dual Euclidean principle

The characteristic feature of stable domains is that they have (and stable functions preserve)pullbacks ie meets of pairs that are bounded above Pullbacks arise in products of domains fromany pair of instances of the order relation for example

(perpgt) 6 (gtgt) (f prime x) 6 (f x)

or or or or

(perpperp) 6 (gtperp) (f prime xprime) 6 (f xprime)

in ΣtimesΣ and in Y X timesX for any f prime 6 f in Y X and xprime 6 x in X The first example says that thereis no stable function Σ times Σ rarr Σ that restricts to the truth table for or The second means thatfor the evaluation map ev Y X timesX rarr Y to be stable

f prime 6 f implies forallxprime x xprime 6 xrArr f prime(xprime) = f prime(x) and f(xprime)

In fact f prime 6 f is given exactly by this formula which is known as the Berry order since (usingthe universal property that defines Y X) the function perp 6 gttimesX rarr Y defined by (perp x) 7rarr f primexand (gt x) 7rarr fx is stable iff the formula holds

Stable domains were introduced by Gerard Berry [Ber78] as a first attempt to capture se-quential algorithms denotationally parallel or with por(tperp) equiv por(perp t) equiv t and por(f f) equiv f is not interpretable as it is in Dcpo Notice that the Berry order is sparser than the pointwiseorder on the function-space it bears some resemblance to the Euclidean principle but I cannotsee what the formal connection might be here or with Berryrsquos domains that carry two differentorder relations

In order that they may be used like Scott domains for recursion stable domains must haveand their functions preserve directed joins with respect to the Berry order Some models alsorequire infinitary (wide) pullbacks [Tay90] ie binary ones and codirected meets The literatureis ambiguous on this point (some such as [FR97] require only the binary form) because thereare also models satisfying Berryrsquos ldquoIrdquo condition that there be only finitely many elements belowany compact element so there are no non-trivial codirected meets to preserve Berry and otherauthors also required distributivity of binary meets over binary joins (the ldquodrdquo condition hencedI-domains) in order to ensure that function-spaces have ordinary joins of bounded sets ratherthan multijoins

From the point of view of illustrating dominances it is useful to assume that stable functionsdo preserve infinitary pullbacks and therefore meets of all connected subsets Then if U sub Xis connected and classified by φ X rarr Σ we may form u equiv

andU isin X and by stability of φ

u belongs to U so U is the principal upper set uarru Removing the connectedness requirementAchim Jung and I picturesquely called the classified subsets (disjoint unions of principal uppersubsets) icicles

The exponential ΣΣ is a V-shape in the Berry order The identity and λxgt are incomparableif there were a link Σrarr ΣΣ between them its exponential transpose would be or Σtimes Σrarr Σ

17

id equiv λx x λxgt

Fminus1(gt)

perp equiv λxperpOn the other hand there is a morphism F ΣΣ rarr Σ for which F (λxgt) equiv perp and F (id) equiv gtshown as the icicle Fminus1(gt) above This control operator detects whether φ isin ΣΣ reads itsargument in a generalised form it is called catch by analogy with the handling of exceptionswhich are thrown When φ is processed sequentially there must be a first thing that it does iteither reads its input or outputs a value irrespectively of the input

Nevertheless ΣΣ is still an internal semilattice carrying the pointwise semilattice order forwhich id 6 (λxgt) and F is not monotone I have not worked out the monadic completion Cof the category of stable domains but it would be interesting to know what this looks like Weconsider the stable example again in Remark 93

5 The Phoa Principle

Remark 51 Consider the lattice dual of the Euclidean principle

σ or F (σ) lArrrArr σ or F (perp)

where we suppress the parameter x isin X to σ isin ΣX and F isin ΣXtimesΣ Taking F to be notnot in Set orPos this yields excluded middle (notnotσ hArr σ) Observe that the Euclidean principle and its dualare trivial for σ equiv gt and σ equiv perp and therefore for the classical case Σ equiv perpgt

In topology and recursion C-morphisms of the form F ΣY rarr ΣX preserve directed joins withrespect to the semilattice order they are said to be Scott-continuous (cf Examples 211)

This completely changes the constructive status of the dual Euclidean principleThe results about open subsets and maps and overt objects that we present later in the paper

then have closed proper or compact mirror images Note that Scott-continuity of notnot would implyexcluded middle

More basically any F ΣY rarr ΣX is monotone (Remark 35) in Pos as well as in topology andrecursion but not in Set Even though we intend to consider (locally compact) topological spacesX in general we need only use lattice or domain theory to study ΣX since this is just the latticeof open subsets of X equipped with a topology that is entirely determined by the (inclusion)order

Lemma 52 For any exponentiating semilattice Σ the functor Σ(minus) is order-enriched iff all func-tions ΣY rarr ΣX are monotone but then it is contravariant with respect to the order if F 6 Gthen ΣG 6 ΣF and if L a R then ΣR a ΣL

Remark 53 In recursion theory ΣX consists of the recursively enumerable subsets of X By theRicendashShapiro theorem [Ric56 Ros86] recursive functions F ΣY rarr ΣX again preserve directedunions The following result has an easier proof in this situation where σ isin Σ measures whether aprogram ever terminates then σ hArr

orn σn where σn decides whether has finished within n steps

or is still running

18

Proposition 54 Suppose that C has stable disjoint coproducts and a dominance Σ that is adistributive lattice Then for every element σ isin Σ there is some directed diagram d I rarrperpgt sub Σ only taking values perp and gt and with I an overt object (Section 7) of which σ isthe join

Proof Intuitively σ is gt just on the part U equiv [σ] of the world that it classifies so σ is thesmallest global element that is above the partial element gt U rarr Σ Since U is classified it isopen Also as it is a subsingleton it satisfies the binary part of the directedness property Weachieve the full property by using d equiv [perpgt] I equiv 1 + U rarr Σ

In more orthodox3 categorical terms σ Γrarr Σ is a generalised element and classifies i U rarrΓ then we put I equiv Γ + U and d equiv [perpgt] Stable disjoint sums in C (Section 9) are needed toshow that d I rarr Σ is directed and I rarr Γ is an open map (Section 7) In fact there a semilatticestructure

I timesΓ I sim= Γ + U + U + Uor - Γ + U equiv I

perpΓ

where U timesΓ U sim= U since U rarr Γ An element τ isin Σ is an upper bound for the diagram d I rarr Σiff its restriction to U is gt as is the case for σ

UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt

If τ classifies V rarr Γ and is a bound then there is a commutative kite so U sub V using the pullbackHowever to deduce φ 6 ψ we need uniqueness of the characteristic maps to Σ or equivalentlythe Euclidean principle

The synthetic form of this argument uses the left adjoint existd ΣΓ times ΣU sim= ΣΓ+U rarr ΣΓ of Σdfor which existd(perpgt) equiv φ Corollary 84 discusses I-indexed joins

[More simply if we formulate the Scott principle cf [E] as

ξ ΣN F ΣΣN ` Fξ lArrrArr exist`KN F (λn n isin `) and foralln isin ` ξn

then the Phoa principle is the special case N equiv 1]

Corollary 55 The dual of the Euclidean principle is therefore also valid in LKLoc By Theo-rem 310 this says that perp isin Σ classifies closed subsets

Proof By Scott continuity and Remark 51 for perpgt

Classically this is trivial but we are making a substantive claim here about how the Sierpinskispace ought to be defined intuitionistically This claim is amply justified by the openndashclosedsymmetry that we shall see in this paper and [D] Indeed Japie Vermeulen has identified the dualFrobenius law for proper maps of locales [Ver94] although he used the opposite of the usual order

3Orthodox though it may be I regard parametric objects like this as unsatisfactory without first defining asystem of dependent types by means of a class of display maps [Tay99 Chapter VIII] The object Γ +U is used fora different purpose in [D Section 7]

19

on a frame so that φ 6 ψ would correspond to inclusion of the closed subsets that they classifyAndre Joyal and Myles Tierney defined the Sierpinski space by means of the free frame on onegenerator [JT84 sectIV 3] and gave a construction in terms of the poset perp 6 gt that amounts tosaying that ΣΣ sim= Σ6

[There is a much simpler proof of the dual Euclidean and Phoa principles for locales in sectO 75]

Corollary 56 By the lattice dual of the results of Section 3 there is a right adjoint Σi a foralli ofthe inverse image map for the inclusion of a closed subset i C - X in LKLoc This satisfiesthe dual of the Frobenius law

foralli(θ) or ψ = foralli(θ or Σiψ)

for any θ isin ΣC and ψ isin ΣX together with BeckndashChevalley again

The same principle is also valid in (parallel) recursion theory where Martin Hyland stressed[Hyl91 Assumption 4] that perp should classify co-RE subsets as well as gt classifying RE subsetsHe also stated the following idea as his Assumption 6 although it was only after writing thatpaper that he attached his former studentrsquos name to it

Proposition 57 Let Σ be an exponentiating object with global elements perp 6 gt in an internalpreorder Then the conjunction of(a) Σ is a distributive lattice

(b) the Euclidean principle σ and F (σ)hArr σ and F (gt)

(c) its lattice dual σ or F (σ)hArr σ or F (perp) and

(d) monotonicity of F with respect to the semilattice orderfor all F X times Σrarr Σ and σ isin Σ is the Phoa principle that

any such F is monotone in this sense and converselyfor each pair of maps φ ψ X rArr Σ with φ 6 ψ pointwisethere is a unique map F X times Σrarr Σ with F (xperp) equiv φ(x) and F (xgt) equiv ψ(x)

In this case F is obtained from φ and ψ by ldquolinear interpolationrdquo Fσ equiv Fperp or (σ and Fgt)Another way of stating the Phoa principle is that 〈evperp evgt〉 ΣΣ rarr ΣtimesΣ is mono and is the

order relation on Σ indeed that for which and is the meet and or the join

Proof [rArr] F must be given by this formula (and so ΣΣ rarr Σtimes Σ is mono) because

Fσ hArr (Fσ or σ) and Fσ hArr (Fperp or σ) and Fσ hArr (Fperp and Fσ) or (σ and Fσ)hArr Fperp or (σ and Fgt)

[lArr] Any function F given by this formula is monotone in σ With X equiv Σ we obtain F equiv and fromφ equiv λxperp and ψ equiv id and F equiv or from φ equiv id and ψ equiv λxgt Now consider the laws for adistributive lattice in increasing order of the number k of variables involved For k equiv 0 we have thefamiliar truth tables Each equation of arity k ge 1 is provable from the ones before when the kthvariable is set to perp or gt so let φ 6 ψ X equiv Σkminus1 rarr Σ be the common values then the two sidesof the equation both restrict to φ and ψ so they are equal for general values of the kth variable byuniqueness of F Σk rarr Σ Finally σandFσ hArr σand

(Fperpor (σandFgt)

)hArr (σandFperp)or (σandσandFgt)hArr

σ and (Fperp or Fgt)hArr σ and Fgt and similarly for the dual

Remark 58 Theorem 42 and Remarks 43(acef) can easily be adapted to show that the Phoaprinciple holds in the free model in which Σ is a distributive lattice either as the lattice dualresult together with a separate proof of monotonicity or by expanding polynomials For the lattermethod the case of application (42(e)) is again the most complicated the induction hypothesisbeing Fperp rArr Fσ hArr (Fperp or σ and Fgt)rArr Fgt [This suffers from the same error as Theorem 42]

20

Definition 59 When all maps F Γtimes Σrarr Σ preserve the semilattice order there are two waysof extending this order to binary relations on any other object X of the category (not just theretracts of powers of Σ as in Remark 35) They both agree with the concrete order in Cont and(assuming excluded middle) Pos(a) The Σ-order defined by

x vΣ y if forallφ isin ΣX φxrArr φy

is reflexive and transitive but not necessarily antisymmetric This is the relation inherited

by X via ηX from the semilattice order on ΣΣX For topological spaces it is known as thespecialisation order [Joh82 sectII 18] but in Set it is discrete (x vΣ y iff x = y)

(b) The link relation is

x vL y if exist`Σrarr X `(perp) = x and `(gt) = y

ie a path from x to y indexed by Σ rather than by the real unit interval as in traditionalhomotopy theory For categories in general this relation need not be transitive or antisym-metric for example it is indiscriminate in Set ie x vL y always holds [assuming excludedmiddle]

Wesley Phoa formulated his principle and introduced the link order to show that the order relationon a limit in his category of domains is given in the expected way [Pho90a sect23] For this to work` must be unique

Remarks 510(a) All morphisms f X rarr Y are monotone with respect to both of the relations that we have

just defined (so when we talk about monotone maps we mean with respect to the semilatticeorder)

(b) The link relation is contained in the Σ-order iff all F Γtimes Σrarr Σ are monotone

(c) In the poset perp 6 gt these two points are in the link relation iff excluded middle holds sincewe need to find a map ` Σrarr perp 6 gt

(d) The Σ-order on ΣX coincides with the semilattice order iff ηΣX is monotone

(e) If Σ is a lattice then the semilattice order on ΣX is contained in the link relation but thePhoa principle makes ` unique

(f) Any replete object X inherits the link relation via ηX X minusrarr ΣΣX so this always happenswhen the adjunction Σ(minus) a Σ(minus) is monadic

(g) When the Phoa principle holds all maps ΣY rarr ΣX are monotone (cf Example 45)

(h) As given these definitions are not internal to the category C they are for generalised elements

(Remark 25) ie in an enclosing topos such as the presheaf topos SetCop

or a model ofsynthetic domain theory One way of translating the definition of vΣ into an internal one is

as the inverse image along ηX of the semilattice order on ΣΣX if the appropriate pullbackexists Similarly vL (with ` unique) can be defined internally as XΣ if this exists

Important though they are in domain theory these order relations will only be mentioned in thispaper in the trivial situation of the following section

6 Discrete and Hausdorff objects

Now we can begin to develop some general topology and logic in terms of the Euclidean and Phoaprinciples and monadicity

21

For the remainder of the paper (apart from the last section) we shall work in a model (CΣ)of the axioms in Remark 38 ie Σ is a Euclidean semilattice and the adjunction is monadic Thecategory could be Set LKLoc CDLatop an elementary topos or a free category as considered inTheorem 42 and Remark 43 We shall also assume the dual Euclidean principle when discussingthe dual concepts closed subspaces compact Hausdorff spaces and proper maps

By Theorem 310 Σ classifies some classM of supports which we call open inclusions So farM has been entirely abstract the only maps that are obliged to belong to it are the isomorphismsThe diagonal map ∆ X rarr X times X is always a split mono so what happens if this is open orclosed

In accordance with our convention about Greek and italic letters (Remark 25) we use p0 X times Y rarr X and p1 X times Y rarr Y instead of the more usual π for product projections though wekeep ∆ for the diagonal

Definition 61 An object X isin obC is said to be discrete if the diagonal X rarr X timesX is open

X - 1

X timesX

∆

cap

(=X)- Σ

gt

The characteristic map (=X) X timesX rarr Σ and its transpose X X rarr ΣX are known as theequality predicate and singleton map respectively We shall often write the subscript on thisextensional (but internal) notion of equality to distinguish it from the intensional (but external)equality of morphisms in the category C

[Symbolically the equality predicate =X is related to equality of morphisms by the rule

a = b X=============(a =X ) lArrrArr gt

in which a = b X is called a statement in Definition I 44]

Lemma 62 If X is discrete in this sense then it is T1 ie the Σ-order (Definition 59(a)) on Xis discrete If all functions Σrarr Σ are monotone then the link order is also discrete

Proof If x vΣ y then x vΣ y in ΣX by Remark 510(a) so by putting φ equiv evx in thedefinition of vΣ by reflexivity we have gt hArr x(x) 6 y(x) equiv (x =X y)

If all F Σrarr Σ are monotone then x vL y ` x vΣ y But for a direct argument (on the samehypothesis) consider F equiv λσ (`σ =X `perp) Then Fperp hArr gt so Fgt hArr gt by monotonicity but thissays that x =X y

Examples 63(a) Every set is discrete

(b) For a poset to be discrete in this sense the diagonal (x y) | x =X y must be an upper subsetof X timesX This means that if x 6 y then (x x) 6 (x y) must also lie in this subset so x = yHence discreteness agrees with standard usage (This is the same argument as in the Lemma)

(c) For a topological space to be discrete the diagonal subset must be open Each singleton xis open so if we may form arbitrary unions of open subsets all subsets are open

(d) In recursion theory N is discrete in the sense of Definition 61 and singletons in N are recur-sively enumerable but arbitrary subsets are not This is explored in [D] Intuitionistically

22

even 1 has co-RE subspaces that are not RE so there is nothing to be gained from introducinga notion of strong discreteness

(e) A presheaf is discrete with respect to Σ equiv Ωj (Example 210(b)) iff it is j-separated [Joh77Proposition 329] [BW85 sect62]

(f) A recursive datatypeX is discrete iff there is a program δ(x y) that terminates iff its argumentsare intensionally equal for example if X is defined by reduction rules or by generators andrelations then δ has to search for an equational proof Following the usage of decidable (yesor no) and semi-decidable (yes or wait) semi-discrete would perhaps be a clearer term

Definition 64 Dually we say that an object is Hausdorff if the diagonal is closed (classifiedby perp) [Bou66 sect8] We write (6=X) or (X) X timesX rarr Σ for the characteristic function whichis sometimes called apartness Again it follows that singletons are closed (the T1 separationproperty in point-set topology) but not arbitrary subsets

The symbolic rule isa = b X

=============(a 6=X ) lArrrArr perp

Exercise 65 To check that you understand how 6=X is defined adapt Lemma 62 to show thatif x vΣ y in a Hausdorff space X then (x 6=X y) hArr perp and explain how it follows from this thatX is T1 in the order-theoretic sense

Examples 66(a) For locales this property is called strong Hausdorffness [Joh82 sectIII 13] but this is because

localic and spatial products are not the same unless we require local compactness as indeedwe do in this paper

(b) An object of a topos is Hausdorff in our sense iff it is notnot-separated and in particular thisalways happens for classical sets Similarly any discrete poset or space whose underlying setis notnot-separable is also Hausdorff the converse also being true for posets Hausdorffness istherefore not a very interesting property for sets and posets and it is better to avoid this termaltogether unless the dual Euclidean and Frobenius principles hold

(c) Following the analogy between open and recursively enumerable subsets a recursive datatypeX is Hausdorff iff there is a program δ(x y) that terminates iff its arguments are unequal(distinguishable) For example the real line R is Hausdorff but not discrete [Theorem I 93]Of course we know this topologically the point is that this is the case computationally asBrouwer tried to remind us in contradiction to the pathological analysis that led to Cantorrsquosset theory and ldquofloating pointrdquo arithmetic in Fortran and other programming languageswhich purport to make equality decidable

(d) In traditional point-set topology and locale theory any T0 group is Hausdorff as is any discretespace but these results depend on being able to form arbitrary unions of open subsets andare therefore not true recursively For example we would otherwise be able to solve the wordproblem for groups ie detect that a group is non-trivial In particular even the singletonsubgroup need not be closed

To sum up discreteness and Hausdorffness are quite different properties

Lemma 67 Let X be discrete and φ isin ΣX Then

exist∆(φ) equiv (λxy (x =X y) and φx) equiv (λxy (x =X y) and φy)

23

Proof We use the Frobenius law exist∆(θandΣ∆ψ) equiv (exist∆θ)andψ with θ equiv gt so (exist∆θ) = (λxy x =X y)Putting either ψ equiv Σp0φ or Σp1φ so ψ = λxy φ(x) or λxy φ(y) we recover φ = Σ∆ψ so the lefthand side of the Frobenius law is exist∆(φ) and the right hand side is one of the other two expressions

[x =X y] cap [φx] sub - [φx]timesX sub - [φx] - 1

Γtimes [x =X y]

cap

sub∆- ΓtimesX timesX

cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt

[x =X y] cap [φy]cup

6

sub - X times [φy]cup

6

sub - [φy]cup

6

- 1

gt

6

Theorem 310 gives an alternative proof using open subsets [(x =X y)andφx] is the same subobjectof ΓtimesX as [(x =X y)and φy] since the composites X rarr X timesX rArr X are equal hence they are thealso same subobject of ΓtimesX timesX so the characteristic maps are equal

Corollary 68 (=X) is reflexive symmetric and transitive

Proof Consider φ equiv λu (y =X u) and φ equiv λu (u =X z)

This is the algebraic characterisation of the equality predicate which we consider in Section 10

Corollary 69(λn φ(n)

)a equiv existn (n =N a) and φ(n)

Here ldquoexistnrdquo is as in Definition 77 So instead of β-reducing the application of a predicate toan argument of type N ie substituting the term a for the variable n throughout the formula φwe can make a local change to the expression-tree and rely on unification to carry out the effectof the substitution [Section A 11]

[Lemma J 59 proves the properties of equality symbolically from the Euclidean principle]

Remark 610 The analogous property for forall in intuitionistic logic is that

forall∆(φ)(x y) equiv (x =X y)rArr φx equiv (x =X y)rArr φy

We need the dual Euclidean and Frobenius principles (Corollary 56) to make this equivalent tothe lattice dual of the Lemma namely

forall∆(φ)(x y) equiv (x 6=X y or φx) equiv (x 6=X y or φy)

for Hausdorff spaces In this case an object that is both discrete and Hausdorff is called decidable cf Proposition 96

Proposition 611(a) 1 is discrete

(b) If Σ has perp then 1 is also Hausdorff

(c) If X and Y are both discrete then so is X times Y

(d) Similarly if they are both Hausdorff assuming that Σ is a distributive lattice

(e) If U sub X is any subset of (ie any mono into) a discrete or Hausdorff object then U is alsodiscrete or Hausdorff

24

Proof [ab] (=1) equiv gt 1times 1rarr Σ and (6=1) equiv perp[cd] (=XtimesY ) equiv (=X) and (=Y ) and (6=XtimesY ) equiv ( 6=X) or (6=Y )

U - X - 1

U times U

- X timesX

cap

- Σ

[e] U rarr X is mono iff the square on the left is a pullback

The next result will be used to prove Theorem 113 so instead of monadicity we assume onlythat Σ is exponentiating but still read the existence of the pullback in Definition 61 as thedefinition of discreteness

Lemma 612 If X is discrete then the maps X X rarr ΣX and ηX X rarr ΣΣX are mono

Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6

Proof If X a = X b then the composites Γ rarr Σ are equal but one of them factorsthrough the pullback so the other does too

Naturality of η with respect to X gives a commutative square

X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66

The map on the right is split mono so by the first part and cancellation ηX is mono

Remark 613 The Lemma helps to explain why there are two ways of defining j-separatedpresheaves (namely being T0 or discrete with respect to Ωj) and the fact that Hausdorffnessimplies sobriety for spaces [Joh82 Lemma II 16(ii)] See also [Joh77 Definition 124] [BW85sect23 Proposition 6] and [Mik76 p 3] concerning the singleton map X in an elementary topos

25

We shall find at the end of the paper that CDLatop satisfies most of our characterisation ofelementary toposes apart from the fact that all sets are discrete Letrsquos briefly explore this analogy

Example 614 Assuming excluded middle the subcategory Set sub Pos sub CDLatop consists ofthe discrete (equivalently Hausdorff) objects

Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ

Set is the reflective subcategory of the Ω-replete objects in Pos just as the sheaf subtopos Ejconsists of the Ωj-replete objects in E [BR98]

The object Ω (the dominance in Set) is the underlying set of Υ (that in Pos) ie its im-age under the right adjoint of the inclusion Set sub Pos The product-preserving left adjoint(components) to the inclusion of categories is the replete reflection

By contrast Ωj is the result of applying the left adjoint sheafification to Ω Also we haveΩ Υ instead of Ωj sub Ω

Remarks 615 In the same way we may ask whether adjoints exist to the inclusions of the fullsubcategories Set and KHaus of (overt) discrete and compact Hausdorff spaces in LKLoc insteadof Pos

Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2

(a) The right adjoint Setlarr LKLoc is the set of points functor

(b) Unfortunately the left adjoint Set larr LKLoc which is the components functor is onlydefined for locally connected locales [BP80 Tay90] but it does preserve products

(c) The underlying set of the Sierpinski space Σ is the subobject classifier Ω and the objects ofthe smaller category are replete overt and discrete with respect to both Σ and Ω

(d) The left adjoint LKLocrarr KHaus is the StonendashCech compactification [Joh82 TheoremIV 21] but it does not preserve finite products

(e) Martın Escardo has shown that the patch topology provides the right adjoint but only tothe inclusion into the category of stably locally compact locales and perfect maps [Esc99]

(f) The patch topology on the Sierpinski space is 2 but 2-replete objects are Stone spaces (totallydisconnected compact Hausdorff spaces) and these do not form a pretopos

The lack of openndashclosed symmetry between these results makes it very unlikely that they have aunifying formulation in our axiomatisation

[The overt discrete objects play the role of sets in ASD If we assume motivated by (a) abovethat the inclusion of the full subcategory of them has a right adjoint U then this subcategory isa topos with Ω equiv UΣ and the whole category is equivalent to that of locally compact locales overthis topos [H]]

Discreteness and Hausdorffness are binary properties relating X to X times X we now turn tothe corresponding nullary ones involving 1 = X0

26

7 Overt and compact objects

[The best introduction to compact and overt subspaces in ASD is that in Sections J 8 and J 11See also [G Section 5]]

The characterisation of open maps in terms of the adjunction existf a Σf in Section 3 (and ofclosed maps using Σf a forallf in Corollary 56) can be generalised to remove the mono requirement

Unfortunately as our category need not have all pullbacks we cannot discuss the BeckndashChevalley condition in the generality that we would like (We do have it sufficiently often forthe purposes of this paper namely to study the full subcategory of overt discrete objects in Sec-tion 10) For this reason we abstain from giving a generally applicable definition of open mapbut as there is no such problem with Frobenius we do make the

Definition 71 Any map f Y rarr X not necessarily mono is called pre-open if Σf ΣX rarr ΣY

has a monotone left adjoint existf satisfying the Frobenius law

existf (ψ) and φ = existf (ψ and Σfφ) for all φ isin ΣX and ψ isin ΣY

where φ and ψ are generalised elements cf Remark 25 and Proposition 82

The origin of the name open is that (for topological spaces) Σf has a left adjoint satisfying

Frobenius iff for every open subobject i U rarr X the image of Uirarr X

frarr Y is open indeed thecharacteristic map of this image is existfφ where φ classifies U However this argument does nothave any meaning for us as we do not yet have any notion of direct image and the one that weshall obtain in Section 10 relies on the present discussion See [Bou66 sect5] for an account of openmaps of spaces and [JT84 Chapter V] for the localic version

Lemma 72(a) All isomorphisms are pre-open maps

(b) Inclusions of open subsets are pre-open maps (Theorem 310)

(c) The composite of two pre-open maps is pre-open

(d) If e X Y is Σ-epi (ie Σe is mono) and f e is a pre-open map then f Y rarr Z is alsopre-open

(e) If m Y rarr Z and ΣΣm are mono and m f is pre-open then f X rarr Y is also pre-open

Proof [andashc] are obvious [d] existf equiv existg middot Σe where g equiv f e [e] Let E equiv Σm middot existg where g equiv m f[the letter E is used to suggest the existential quantifier here it does not denote a nucleus] Thenwe easily have φ 6 Σg middot existgφ = Σf middotΣm middot existgφ = Σf middotEφ Using the hypothesis that Σm is Σ-epi forthe other two properties it suffices to consider ψ = Σmθ Then

E middot Σf middot Σmθ = E middot Σgθ = Σm middot existg middot Σgθ 6 Σmθ

and E(φ and Σf middot Σmθ) = Σm middot existg(φ and Σgθ) = Σm(existgφ and θ) = Eφ and Σmθ

Definition 73 Similarly any map f not necessarily mono for which Σf a forallf exists and satisfiesthe dual Frobenius law is called pre-proper Again a continuous function between spaces orlocales is pre-proper iff the image of every closed subset of X is closed in Y See [Bou66 sectsect5 10]for the theory of proper maps of spaces and [Ver94] for locales and the dual Frobenius law inparticular

Closed subsets proper and pre-proper maps satisfy the analogue of Lemma 72

Remark 74 The BeckndashChevalley condition (Propositions 311 and 81) is automatic for (pre-)open maps of spaces and locales but not for pre-proper maps In view of the strict duality between

27

them in our theory this difference in the traditional ones means that we cannot get the BeckndashChevalley condition for free in either case In keeping with this duality it seems inappropriate toemploy the usual name closed for pre-proper maps

Remark 75 Andre Joyal and Myles Tierney do construct pullbacks of open maps of localesagainst arbitrary maps and prove the BeckndashChevalley condition [JT84 Proposition V 41] Butthey do this with the benefit of a development of ldquolinear algebrardquo for sup-lattices which are toAbelian groups as frames (locales as they call them) are to commutative rings [opcit Chapter I]In particular the required pullback of spaces is a pushout of frames and is constructed as a tensorproduct of sup-lattices which is obtained as a coequaliser Our categories do not have arbitrarycoequalisers though it seems plausible that the one that is needed could be constructed Clearlywe are currently even less equipped to undertake an analysis of descent parallel to theirs

We shall concentrate on the question of whether product projections are open or proper andon open maps between overt discrete spaces in Section 10

Remark 76 The openndashproper symmetry brings us to the question of why we have three wordsclosed proper and compact (not to mention perfect) in one case and only open in the otherWithout them of course there would ambiguity over closed but non-compact subsets of non-compact spaces (Proposition 83) But open sets are equally ambiguous [Indeed we find thatovert subspace in real analysis are often also closed cf Definition J 111]

Hence the introduction of the word4 overt for objects keeping open for the subsets and maps

Definition 77 An object X isin obC is said to be overt if Σ has a monotone left adjoint existX ΣX rarr Σ and compact if there is a monotone right adjoint forallX The Frobenius laws are automatic(Proposition 82)

We write existx φ(x) and forallx φ(x) for existX(φ) and forallX(φ) where φ isin ΣX Extending the notationalconvention in Remark 25 the range (X) of such a quantifier must be an overt or compact objectrespectively whilst the type of the body φ like that of a λ-abstraction must be a power of Σ orthe carrier of an algebra

Examples 78(a) Every set presheaf or poset is both overt and compact

(b) Classically every domain topological space or locale is overt

(c) In recursion N is overt as are all recursively enumerable datatypes

(d) See [JT84 sectV 3] and [Pho90a sect65] for some discussion of overt objects in particular thepartial-function space [N X]

(e) If every function ΣY rarr ΣX is monotone then existX equiv evgt a Σ a forallX equiv evperp for any object Xthat has gt and perp by Lemma 52 because perp a a gt

(f) In particular every domain (with perp) is compact

(g) Similarly all stable domains are compact since the only icicle to which perp belongs is the wholedomain (Example 45)

4Unfortunately this distinction cannot be translated into (for example) French but whilst overt obviously camefrom French it has been recorded in English at least since 1330 it means public or up-front This seems to beappropriate for a concept thatrsquos related to having a definite distinction between termination and divergence orbetween habitation and emptiness The etymology also parallels our openndashclosed symmetry in that the changefrom aperıre to lowastoperıre in regional Vulgar Latin was influenced by lowastcoperıre from which we get cover and covert[Bar88]

28

But there are also other compact predomains ie not necessarily having perp This intriguingpossibility is thrown up by our unification of topology and recursion future work will uncovertheir significance

[Since the publication of this paper Martın Escardo [Esc04] has written extensively on thecomputational significance of this notion of compactness]

Remark 79 The usual definition of compactness for topological spaces that every cover by opensubsets has a finite subcover can be reformulated in terms of directed joins cf [Joh82 sectIII 1]for locales Our notion of compactness (in the diagram on the right below) is equivalent to theusual one for LKSp and LKLoc because forallX must be a map in the category and is thereforeScott-continuous (Examples 211 Remark 711)

Proposition 710 If the quantifiers and perp exist then they must form pullbacks as shown

1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt

Conversely if gt 1rarr ΣX is open then its classifier is forallX Likewise assuming the dual Euclidean principle if perp 1 rarr ΣX is closed then its classifier is

existX

Proof existXφhArr perp iff φ = λxperp and forallXφhArr gt iff φ = λxgtConversely if gt sub ΣX is classified by (some map that we call) forallX then we need to show

that forallX is monotone id 6 forallX middot Σ and Σ middot forallX 6 id

U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt

Given φ ψ isin ΣX with φ 6 ψ let U V sub Γ be their pullbacks against gt 1 rarr ΣX which existbecause forallXφ and forallXψ classify them Then ψ iU gt φ iU = gt whence U sub V so forallXφ 6 forallXψby uniqueness of classifiers

1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29

Both of these squares commute but the one with id is a pullback so comparing the other withthe pullback that it contains we have

1 equiv [id] sub [forallX middot Σ]

and so the required inequality follows by uniqueness of classifiers

X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap

forallX times id- ΣtimesX

(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt

The two lower composites are the exponential transposes of Σ middot forallX and id respectively and bothdiagrams are pullbacks So to obtain the required inequality (again using uniqueness of classifiers)we only need to check that one subset is contained in the other but clearly φ[x] = gt whenφ = λxgt (We have equality when X is inhabited but not when itrsquos empty)

The analogous result for existX depends on the dual Euclidean principle since we rely on unique-ness of classifiers using perp

Remark 711 The simplest way of imposing Scott-continuity is the equation

F (λxN gt) lArrrArr existnN F (λxN x lt n) for all F isin ΣΣN

which was called the Scott Principle in [Tay91] In this situation N cannot be compact becauseF = forallN would satisfy

forallN(λxgt) equiv (forallxN gt)hArr gt forallN(λx x lt n) equiv (forallxN x lt n)hArr perp

making the two sides of the Scott principle gt and (existnperp)hArr perp

Whilst Scott continuity pervades the motivations of Abstract Stone Duality it is remarkablehow many theorems we can prove before we need to invoke it as an axiom In particular the searchor minimalisation operator micro (2perp)N rarr Nperp for general recursion can be constructed without usingit [D] but we do need it for the function-space (Nperp)N [F] For all of the investigations that I havedone so far in general topology the Phoa principle and monadicity have been enough

Proposition 712 If X is overt and discrete (in particular X equiv N) then X X ΣX is a

Σ-split mono ie there is a map I ΣX ΣΣX such that Σ X middot I = idΣX

X- X - ΣX

ΣXλx F (λy x =X y)larr7 F-

φ 7rarr λψ existy φy and ψy- ΣΣX

This subspace is in general neither open nor closed but N sub - Nperp - ΣN assuming the Scottprinciple [D F]

30

8 The quantifiers

Having used the symbols exist and forall we are obliged to justify them in terms of the rules of naturaldeduction or at least their categorical interpretation [Tay99 sectsect93ndash4] as we donrsquot want to get tooheavily involved in syntax In particular the ability to substitute under a quantifier is anotherconsequence of insisting that the adjunctions be internal

[The letter E is used in the following two results to suggest an existential quantifier not anucleus]

Proposition 81 If Σ Σ rarr ΣX has a left (or right) adjoint then Σp1 ΣY rarr ΣXtimesY also hasone and this automatically satisfies the BeckndashChevalley condition the pullbacks in question beinggiven by product projections as shown on the left below

X times ZX times f- X times Y

p0- X ΣXtimesZ ΣXtimesf

ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6

Proof If E a Σ then F equiv EY a Σp1 by Lemma 36Explicitly for ω isin ΣXtimesY put F (ω) equiv λy E

(λx ω(x y)

) then for ψ isin ΣY

F(Σp1(ψ)

)= λy E(λx ψy) = λy EΣ(ψy) 6 λy ψy = ψ

whilst ω(x y)rArr ΣE(λxprime ω(xprime y)

)hArr F (Σp1ω)(y)

The BeckndashChevalley condition is naturality of E(minus) ΣXtimes(minus) rarr Σ(minus) with respect to f Explicitly ω 7rarr λz E

(λx ω(x fz)

)both ways round the square involving F and G equiv EZ

The preceding proof only required Σ to be exponentiating the Euclidean principle comes intothe next result

Proposition 82 For X overt every product projection p1 X times Y rarr Y is pre-open because(a) by the Euclidean principle X rarr 1 is pre-open ie existX ΣX rarr Σ obeys the Frobenius law

(cf [JT84 Proposition V 3 1] for locales)

(b) for any pre-open map f X rarr Z the map f times Y X times Y rarr Z times Y is also pre-open(cf Bourbakirsquos definition of proper maps [Bou66 sect10])

Proof(a) For σ isin Σ and φ isin ΣX let F (σ) equiv E

(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6

Definition 77 required E to be monotone so F (σ)rArr E(Σσ)rArr σ Then

E(φ and Σσ)hArr F (σ)hArr F (σ) and σ hArr F (gt) and σ hArr E(φ) and σ

31

by the Euclidean principle

(b) Let φ isin ΣXtimesZ be a generalised element in the context Γ so φ Γrarr ΣXtimesZ then φprime equiv λx φ(x z)is a generalised element in the context Γ times Z and this is how Definition 71 must be readFrom Proposition 81 existftimesZφ = λz existf

(λx φ(x z)

) Then for ψ isin ΣYtimesZ

existftimesZ(φ and ΣftimesZψ) = λz existf(λx φ(x z) and ψ(fx z)

)= λz

(existf (λx φ(x z)) and λy ψ(fx z)

)= existftimesZφ and ψ

Proposition 83 By analogy with Proposition 611(a) 1 is overt and compact

(b) If X and Y are both overt or both compact then so is X times Y

(c) If U rarr X is an open subset of an overt object then U is itself an overt object

(d) Similarly for closed subsets of compact objects

Proof [a] exist1 equiv forall1 equiv idΣ [b] existXtimesY equiv existX middot existXY equiv existY middot existYX [cd] U rarr X rarr 1 is a composite ofopen or proper maps

[Beware that the letter I in the following two results denotes an ldquoindexing objectrdquo not aΣ-splitting]

Corollary 84 If the object I is overt then every algebra (A equiv ΣX) has internal I-indexed joinsand distributes over them and they are preserved by homomorphisms

X times If times I- Y times I AI equiv ΣXtimesI

ΣftimesIBI equiv ΣYtimesI

X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI

Dually if I is compact then algebras have and homomorphisms preserve I-indexed meets

Proof These are re-statements of the Frobenius and BeckndashChevalley conditions

Corollary 85(a) In classical topology every object I is overt Therefore algebras have all joins and binary

meet distributes over them ie the algebras are frames and the homomorphisms preservejoins By the adjoint function theorem Heyting implication exists in the algebras and framehomomorphisms have (non-continuous) right adjoints

(b) In recursion N is overt but other objects need not be The algebras are sometimes calledσ-frames

(c) N is not compact so N-indexed meets need not be preserved

(d) If 0 and 2 are overt then each algebra is an internal distributive lattice which we shall considerin the next section

32

Now letrsquos think a bit about syntax using [Tay99 sect93]

Remark 86 Our category is a model of a fragment of predicate calculus in which each objectnames a (non-dependent) type and contexts are products (cf Theorem 42) Each open inclusionU rarr X is a predicate x X ` φ(x) prop though we prefer to regard φ(x) as a generalised elementof ΣX rather than as a mono Thus we interpret

Γ ` φ prop by φ isin Σ ie φ Γrarr Σ and

Γ φ1 φ2 φn ` θ by φ1 and φ2 and middot middot middot and φn rArr θ isin Σ

The effect of pullback Σp0 equiv plowast0 along a product projection p0 ΓtimesX rarr Γ is to add a variable xto the context (weakening which is written xlowast in [Tay99])

Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ

If X is overt then this map Σp0 has a left adjoint existX ΣX rarr Σ or existp0 ΣΓtimesX rarr ΣΓ which

interprets existential quantification

Γ x X ` φ(x) prop

Γ ` existx φ(x) prop

Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

The BeckndashChevalley condition is needed to ensure that for any function f Γrarr ∆ between typesthe bijective correspondence on the right is preserved by Σf (substitution or cut along f) whilstthe Frobenius law provides in a similar way for additional predicates that may be present in thecontext

Remark 87 In Section 10 we shall encounter expressions of the forms

existi middot Σi middot Σp0 existp1middot existi middot Σi middot Σp1 and existp1

middot existi middot Σi middot Σp0

where i R rarr X times Y is the inclusion of an open binary relation classified by ρ X times Y rarr ΣRecall from Section 3 that existi middot Σi equiv ρ and (minus) so it takes

Γ x X y Y ` ψ(x y) to Γ x X y Y ` ρ(x y) and ψ(x y)

without changing the context Hence the effect on Γ x X ` φ(x) prop of

existi middot Σi middot Σp0 is Γ x X y Y ` ρ(x y) and φ(x) prop

existp1middot existi middot Σi middot Σp1 is Γ y Y ` existxX ρ(x y) and φ(y) prop

existp1middot existi middot Σi middot Σp0 is Γ y Y ` existxX ρ(x y) and φ(x) prop

This brief discussion of the rules of natural deduction and Corollary 1011 about the directimage show logicians and categorists respectively that we are using the existential quantifier in theusual way However the dual of the Euclidean principle implies the dual Frobenius law for forallX which is something extra on top of the standard rules for the universal quantifier [Tay99 sect94]namely that Σ a forallX with the BeckndashChevalley condition

Remark 88 Let φ isin ΣX be a decidable predicate on any overt compact object so forallx(φ(x) or

ψ(x)) where ψ equiv notφ isin ΣX Then we have forallx

(φ(x) or existy ψ(y)

) which is equivalent by the dual

Frobenius law to(forallx φ(x)

)or(existy ψ(y)

)

33

For decidable predicates on N this property is well known in recursion theory as the Markovprinciple [Mar47] [Ros86 sect51]

For us it is not legitimate to write ldquoforallnN φ(n)rdquo because N is not compact in topology orrecursion (Remark 711) But consider its one-point compactification Ninfin both N and Ninfin are(intended to be) overt and the inclusion i N rarr Ninfin is dense in the sense that

(existxNinfin ψ(x)

)hArr(

existnN ψ(in)) Although N is not compact if gt hArr forallxNinfin φ(x) isin Σ then Σiφ = gt isin ΣN

Conversely we may transfer Σ-predicates from N to Ninfin but we cannot unfortunately do thesame with decidable ones without prejudging the question for the extra point infin isin X

We have a more encouraging result when we compare our universal quantifier for closed sub-sets with the standard one for all subsets in a topos The following situation arises in syntheticdomain theory (Remark 213) for the sheaves for a LawverendashTierney topology (Σ equiv Ωj in Exam-ple 210(b)) and also for Ω Υ in Example 614 since this map is actually also mono

[Eduardo Dubuc and Jacques Penon [DP86] called an object K of a topos compact if itsuniversal quantifier forallK satisfies the dual Frobenius law]

Proposition 89 Suppose that Σ is an object that satisfies the Euclidean principle in a topos soΣ classifies certain open subsets whilst Ω classifies all subsets If the object X is compact withrespect to Σ then the interpretations of forallX with respect to Σ and Ω agree in the sense that thebottom face of the cube commutes

Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-

Proof To prove that the two routes ΣX rarr Ω are equal it is enough to show that they bothclassify the Ω-subobject λxgt sub ΣX The front and back faces of the cube are pullbacks byProposition 710 as is the right face because Σ rarr Ω is mono by Remark 213 To show that theleft face is also a pullback consider any test Γ the two routes ΓrArr ΩX are equal iff the square onthe right commutes which it does iff φ = λxgt as required

Remark 810 Of the other meanings that we might attribute to saying that the two notions of forallagree one is trivially true in that ΣX sim= ΣX where X is the replete reflection of X and another istrivially false in that every object is compact with respect to Ω but not necessarily with respectto Σ

Remark 811 It is not possible to adapt this argument to exist and perp because without excludedmiddle characteristic maps with respect to perp 1rarr Ω in a topos need not be unique An analogousresult can nevertheless be achieved by imposing the open cover LawverendashTierney topology on thetopos but discussion of this relies on sheaf-theoretic methods which are inappropriate for thispaper

34

9 Unions and coproducts

By the monadicity property our category has finite coproducts indeed ΣX+Y sim= ΣX times ΣY Inthis section we consider the question of whether these coproducts are stable and disjoint andinvestigate the consequences of assuming that particular objects are overt

[Theorem B 118 constructs coproducts of general spaces in the category and shows that theyare stable and disjoint just assuming that Σ has a point]

Proposition 91(a) The initial object 0 is overt iff Σ has a least element perp Similarly the existence of gt means

that 0 is compact The Frobenius laws say that σ and perp hArr perp and σ or gt hArr gt which alwayshold in a lattice

(b) 2 is overt iff Σ has binary joins or and (the Frobenius law for exist2 says that) and distributes overthem Similarly the existence of and means that 2 is compact distributivity again being thedual Frobenius law (for forall2)

(c) If 0 is strict then it is both discrete and Hausdorff because ∆ 0 sim= 0 times 0 is classified withrespect to both gt and perp by the unique map 0times 0rarr Σ

(d) If + is disjoint and times distributes over it then 2 is discrete and Hausdorff

Proof [b] is Corollary 84 To see α and (β or γ) hArr (α and β) or (α and γ) directly from the Euclideanprinciple consider F (σ) equiv (σ and β) or (σ and γ) [d] is Proposition 95 below

So for this section we shall assume that Σ is a distributive lattice We shall not use the dualEuclidean principle or monotonicity so the results are applicable to elementary toposes The mainone says that in the coproduct of spaces the two components are embedded as complementaryopen subsets The coproduct is therefore stable and disjoint and the empty space is strict

Theorem 92 The category C is extensive ie it has stable disjoint coproducts [Coc93 CLW93][Tay99 sect55] cf [JT84 Corollary to V 2 1] for locales

X - Z Y

1

- 2

s

1

Proof Given any commutative diagram in C as shown above we must show that the top row isa coproduct of spaces (Z = X + Y ) iff the squares are pullbacks (inverse images) We do this byconsidering the corresponding diagram of algebras with A equiv ΣX B equiv ΣY and C equiv ΣZ

First let φ equiv Σs(gtperp) ψ equiv Σs(perpgt) isin C Since Σ is a lattice and Σs is a homomorphismφ and ψ hArr perp and φ or ψ hArr gt in C The definition of φ makes the diagram

X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35

commute in which the square rooted at 2 is a pullback iff that at Σ is oneNow suppose that Z = X+Y so C = AtimesB Then φ equiv (gtperp) isin AtimesB and C darr φ equiv (AtimesB) darr

(gtperp) sim= A This means that X sim= [φ] ie the square rooted at Σ is a pullback whence so is thatat 2

Conversely suppose that the two squares are pullbacks so X = [φ] and Y = [ψ] Then

AtimesB sim= (C darr φ)times (C darr ψ) sim= (σ τ) | σ 6 φ amp τ 6 ψσ τ 7rarr σ or τ-θ and φ θ and ψ larr7 θ

C

is an isomorphism because using distributivity

(θ and φ) or (θ and ψ) = θ and (φ or ψ) = θ and gt = θ

(σ or τ) and φ = (σ and φ) or (τ and φ) = σ or perp = σ

since σ 6 φ and τ and φ 6 ψ and φ = perp

Remark 93 The category of stable predomains (ie of disjoint unions of stable domains Exam-ple 45) is also extensive because the forgetful functor to Set creates coproducts and pullbacks(in the category not the domains) We may also see this by a version of the preceding argumentsince it only depends on being able to define φ or ψ when φ and ψ are disjoint (φ and ψ = perp) interms of the systems of icicles that they classify to construct φ or ψ each φ-icicle must be eitherwholly contained in a single ψ-icicle or wholly disjoint from them and vice versa NeverthelessΣ is not an internal lattice mdash even classically where it has only two points

Proposition 94 If X and Y are both overt or both compact then so is X + Y If f1 X1 rarr Y1 and f2 X2 rarr Y2 are both pre-open or both pre-proper maps then so is

f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define

existX+Y ΣX+Y sim= ΣX times ΣY rarr Σtimes Σrarr Σ by (φ ψ) 7rarr (existx φx) or (existy ψy)

forallX+Y by (forallx φx) and (forally ψy) and existf1+f2 by existf1 times existf2 ΣX1 times ΣX2 rarr ΣY1 times ΣY2 The adjunction and Frobenius laws hold componentwise

Proposition 95 If X and Y are both discrete or both Hausdorff then so is X + Y

Proof The decomposition in the diagram depends on distributivity but to recover X+Y as thefourfold coproduct [=X ] + [perp] + [perp] + [=Y ] also requires that coproducts be stable and disjoint

X + Y - 1

(X + Y )times (X + Y )

cap

sim= X2 +X times Y + Y timesX + Y 2 [(=X)perpperp (=Y )]- Σ

gt

Similarly ( 6=X+Y ) is given by [6=X gtgt 6=Y ]

Proposition 96 Assuming the dual Euclidean principle any subset U that is both open andclosed is a component of a disjoint union as in Theorem 92

36

Proof Let U equiv φminus1(gt) = ψminus1(perp) and put V equiv φminus1(perp) and W equiv ψminus1(gt) so

0 = U capW = (φ and ψ)minus1(gt) 0 = V cap U = (φ or ψ)minus1(perp)

By uniqueness of characteristic maps of both kinds φ and ψ = perp and φ or ψ = gt so we have thesituation of Theorem 92 V equivW being the complement of U

In a non-Boolean topos by contrast a subset that is both open and closed in our sense neednot be complemented but merely notnot-closed

10 Open discrete equivalence relations

From now on we concentrate on the full subcategory of overt discrete objects showing that it is apretopos The notion of pretopos is the finitary part of Jean Giraudrsquos categorical characterisationof Grothendieck toposes [Joh77 Theorem 045] These are the properties of the category of ldquosetsrdquothat we require in order to do algebra and symbolic logic in it for accounts of which see [MR77][FS90] [Tay99 Chapter V] In particular we shall show how to construct quotients by equivalencerelations using a Σ-split coequaliser

For point-set topology related results in both the open and proper cases are to be found in[Bou66 sect52] except that there an open equivalence relation is by definition one for which q is anopen map

As we do not currently have pullbacks of general open maps (Remark 75) we cannot yetdevelop relative versions of these results for etale maps D rarr Z which Joyal and Tierney defineas open maps for which ∆ D rarr D timesZ D is also open [JT84 sectV 5]

Proposition 101 Pullbacks rooted at any discrete object D exist

P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i

cap idtimes g- X timesD

(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37

Lemma 102 If X is an overt object then f and glowastf are pre-open maps and the BeckndashChevalleycondition holds

Proof (id f) X rarr XtimesD and p1 XtimesD rarr D are pre-open maps (Propositions 311 and 82)where for φ isin ΣX exist(idf)φ equiv λxd φ(x) and (fx =D d) so

existfφ = λd existx φ(x) and (fx =D d)

Corollary 103 If i X rarr D is a mono between overt discrete objects then X is classified bysome φ isin ΣD

Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6

Proof The square on the left is a pullback iff i is mono and then the BeckndashChevalley condition(Propositions 311 and 81) makes the square on the right commute (Σi middot existi = id) which was thecondition required in Theorem 310 [See sectO 810 for a simpler symbolic proof]

Definition 104 A pre-open map f is surjective if id = existf middot Σf This is equivalent to (existfgt)hArr gt by the Frobenius law cf [JT84 sectV 4]

Lemma 105 Let f be a pre-open surjective map [from an overt object to a discrete one] Thenglowastf is also pre-open surjective

Proof By the proof of Lemma 102 surjectivity of f X rarr D says that

λdgt = existfΣf (λdgt) = existf (λxgt) = λd existx(fx =D d)

which means ldquoforalld isin D existx isin X (fx =D d)rdquo externally cf Remark 88We require existX middot existi middotΣi middotΣp1 = id where the effect of existi middotΣi was described in Remark 87 This

is the case for ψ isin ΣY because(existx ψyand (fx =D gy)

)hArr (ψyandexistx fx =D gy)hArr ψy by Frobenius

and surjectivity of f at d equiv gy

Proposition 106 If X and Y are both overt or both discrete then so is P

Proof existP equiv existY middot existglowastf and (=P ) equiv (=X) and (=Y ) cf Propositions 611 and 83

Proposition 107 For any morphism f X rarr D from an overt object to a discrete one thekernel pair K rArr X of f exists and i K rarr X times X is an open equivalence relation (reflexivesymmetric and transitive)

38

Lemma 108 Let X be an overt object and i K rarr XtimesX an open equivalence relation classifiedby δ X times X rarr Σ Then the coequaliser K rArr X Q exists in C and X Q is a pre-opensurjection [See Example B 1113 for a construction using a nucleus in the ASD λ-calculus Infact K can be also constructed as a subspace of ΣX ]

∆

Q q

X p0

p1

K

X timesX

i

p1

p0

Proof Write p0 equiv p0 i and p1 equiv p1 i

Since the monadic property says that Σ(minus) C Algop we calculate the coequaliser q ofp0 p1 K rArr X as the equaliser Σq of the homomorphisms Σp0 and Σp1 in Alg As monadicforgetful functors create equalisers it suffices to show that the carrier of this equaliser exists as anobject of C when we just consider Σp0 and Σp1 as functions (C-morphisms) To do this we showthat existp1 splits the equaliser ie that ΣQ is a retract of ΣX not just a subobject

The equations to be verified are

existp1 middot Σp1 = idΣX and Σp0 middot existp1 middot Σp0 = Σp1 middot existp1 middot Σp0

cf Lemma 33 They make Q a Σ-split coequaliser First existp1 middotΣp1 = existp1 middot existi middotΣi middotΣp1 takes φ to

(λy existx φ(y)and δ(x y)

)= λy φ(y) by Remark 87

and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-

For the other equation that the two composites Σp01 middot existp1 middotΣp0 (for 0 and 1) are equal it sufficesto post-compose the mono existi and show that

existi middot Σi middot Σp01 middot existp1 middot existi middot Σi middot Σp0

are equal By Remark 87 these composites take φ isin ΣX to

λxy δ(x y) and existz(δ(x z) and φ(z)

)and λxy δ(x y) and existz

(δ(y z) and φ(z)

)respectively These are indeed equal by symmetry transitivity and the Frobenius law

Since U is defined to split the idempotent existp1middot Σp0 we have existq middot Σq = idU and Σq is a

homomorphism by Beckrsquos theorem Hence

Σq middot existq = existp1middot Σp0 = existp1


39

which takes φ isin ΣX toλx existy δ(x y) and φ(y)

by Remark 87 Hence φ 6 Σqexistqφ by reflexivity so existq a ΣqFor the Frobenius law again it suffices to apply the mono Σq which preserves and Let ω isin ΣQ

and put ψ equiv Σqω so ω = existqΣqω = existqψ and ψ = Σqexistqψ Then

Σqexistq(φ and Σqω) which is λx existy δ(x y) and φ(y) and ψ(y) and

Σqexistqφ and Σqω which is λx existy δ(x y) and φ(y) and ψ(x)

are the same by symmetry and transitivity

Lemma 109 Q is discrete and effective

K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --

Proof The diagram commutes by manipulation of products and because q p0 = q p1 We havejust shown that q is pre-open whence so are qtimesX etc by Proposition 82 whilst i is pre-open byhypothesis So the composite K rarr QtimesQ is pre-open but K Q is Σ-epi so ∆ is also pre-openby Lemma 72 ie exist∆ a Σ∆ satisfies Frobenius Hence the split mono ∆ is the inclusion of anopen subset by Theorem 310 ie Q is discrete Using surjectivity of K Q

exist∆ middot Σ∆ = exist∆ middot existqp0middot Σqp0 middot Σ∆ = existqtimesq middot existi middot Σi middot Σqtimesq = existqtimesq

(δ and Σqtimesq(minus)

)= existqtimesq(δ) and (minus)

by Frobenius for existqtimesq so the characteristic map of ∆ is (=Q) equiv existqtimesq(δ)Then Σqtimesq(=Q) equiv δ since q times q is surjective ie qx =Q qy hArr δ(x y) which says that the

quotient is effective

Theorem 1010 The full subcategory of overt discrete objects is effective regular and is a pretoposif Σ is a distributive lattice

Proof(a) Finite limits exist by Propositions 611 83 101 and 106

(b) effective quotients of equivalence relations exist by Lemmas 108 and 109

(c) they are stable under pullback by Lemma 105

(d) coproducts are stable and disjoint and the initial object is strict by Theorem 92

Corollary 1011 Every map between overt discrete objects factorises as an open surjection fol-lowed by an open inclusion This factorisation is unique up to unique isomorphism and stableunder pullback along arbitrary C-maps

Proof Given f X rarr Y form the quotient q X Q of the kernel pair K rArr X of f Thenthe mediator i Qrarr Y is mono [Tay99 sect58] and open by Corollary 103

40

Remark 1012 Although ΣX isnrsquot discrete (except in a topos) Q is also the image factorisationof δ X rarr ΣX

Xq -- Q-

Q- ΣQ-Σq- ΣX

where we check that the composite takes x to λy (qx =Q qy) which is λy δ(x y) by effectivenessThe surjection is Σ-split as are the inclusions by Proposition 712

This is the traditional construction of the quotient as the set of equivalence classes an elementof Q can be represented either by any element of X that is in the equivalence class or by thecharacteristic function of this class The quotient is also constructed using this image factorisationin [Joh77 Proposition 123] see also [BW85 sect23 Theorem 7]

Peter Freyd and Andre Scedrov [FS90] have shown how to capture the notions of effectiveregular category and pretopos in terms of relations instead of functions This approach alsotransfers attention away from objects and on to the morphisms so it is possible for there to beldquotoo fewrdquo objects for the logic their condition of tabulation says that all of the objects thatthe logic describes are actually present Tabulation plays an analogous role in their theory to themonadicity property in ours and to the axiom of comprehension in set theory

See [CW87] for another categorical account of relations

Proposition 1013 Assuming only the Euclidean law and not monadicity the overt discreteobjects of C carry the structure of a (C-enriched) allegory

Proof The hom-set Rel(XY ) is ΣXtimesY which is an internal semilattice The identity on thediscrete object X is (=X) and the composition ΣXtimesY times ΣYtimesZ rarr ΣXtimesZ at the overt object Y isdefined by

σ ρ = λxz existy ρ(x y) and σ(y z)

The unit law is that σ(x z) hArr existy (x =X y) and σ(y z) and associativity follows as usual from theFrobenius law which itself comes from the Euclidean principle (Theorem 310) For the other twoFreydndashScedrov axioms we have

(σ and τ) ρ = λxz existy ρ(x y) and(σ(y z) and τ(y z)

)6 (σ ρ and τ ρ) = λxz

(existy ρ(x y) and σ(y z)

)and(existyprime ρ(x yprime) and τ(yprime z)

)(σ ρ and τ) = λxz


)and τ(x z)

6 σ (ρ and σop τ) = λxz existy ρ(x y) and(existzprime τ(x zprime) and σ(y zprime)

)and σ(y z)

which follow from the Frobenius law and the adjunction existY a Σ by putting yprime equiv y and zprime equiv z

Proposition 1014 If the monadic property also holds then this allegory is tabular and istherefore equivalent to the category of relations of a regular category

Proof Given a relation ρ XtimesY rarr Σ we must find the corresponding open subset U sub XtimesY Lemma 39 did this

Remark 1015 As usual similar results for compact Hausdorff spaces follow from the dualEuclidean principle in particular they too form a pretopos

The root of the distinction between the properties of overt discrete and compact Hausdorffspaces is that N is overt discrete and Hausdorff but not compact (Remark 711) From Corol-lary 84 it follows that all homomorphisms preserve N-indexed joins (but not necessarily meets)

41

whilst existf and forallf where they exist preserve joins and meets respectively by virtue of beingadjoints

Remark 1016 This opens the way to applying the limitndashcolimit coincidence from domaintheory [Tay87] to the construction of infinitary colimits of overt discrete spaces and limits ofcompact Hausdorff ones The following remarks are only intended to sketch the argument as thequestions of the existence of the relevant limits of algebras in C and the internal language neededto invoke them are outside the scope of this paper

A (filtered) colimit diagram of overt discrete spaces is given by a limit of the correspondingalgebras and maps of the form Σf but this is accompanied by the diagram of the left adjoints existf As all of these maps preserve N-indexed joins the limit and colimit coincide Then the limitingcone and colimiting cocone consist respectively of the inverse images and quantifiers for the colimitof the original diagram of overt discrete spaces In particular the colimit is overt and discreteThe subcategory also admits initial algebras [E] so it is an arithmetic universe as defined byAndre Joyal

Similarly a (cofiltered) limit diagram of compact Hausdorff spaces gives rise to a filtered colimitof inverse image maps that coincides with the limit of their universal quantifiers This subcategoryalso admits final coalgebras

11 Monadicity for elementary toposes

This section characterises the case where E is an elementary topos and Ω its subobject classifier[Joh77 Chapter 1] [BW85 Chapter 2] We change the notation from (CΣ) to (E Ω) to emphasisethat this is the only section of the paper in which we either assume directly that E is a topos (andΩ classifies all monos) or make other assumptions that turn out to be equivalent to this

In the earlier parts of this paper we have tried to generalise as much as possible of the basictheory of elementary toposes from higher order to geometric logic ie from Set and toposes toLKLoc and abstract Stone duality We begin this section with a proof of Parersquos theorem Thereason for doing this is to show what (little) remains that is peculiar to the topos case andapparently cannot be generalised [Equideductive topology also incorporates the ideas of Lemmas119 and 1111 below into topology so even less remains that is peculiar to the topos case] Somefurther simplifications could be made with the aid of the theory of replete objects [BR98] and weswitch notation back to Σ for some parts of the present argument that can easily be generalised

We conclude with a ldquoconverserdquo of Parersquos theorem a new characterisation of elementary toposesthat is not based on the notion of subset

Definition 111 An elementary topos is a category E with an exponentiating classifier Ω as inSection 2 but for all monos In particular since X rarr XtimesX is classified all objects are discreteso all equalisers and pullbacks exist by Proposition 101

Bill Lawvere and Myles Tierney originally included finite limits and cartesian closure in thedefinition [Law71] but these conditions are redundant

42

Remark 112 Anders Kock [KM74] constructed function-types by applying (minus)Y

to the pullbacksquare on the left

X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt

where X is mono by Lemma 612 and sX(φ) says that φ is a singleton5 Appealing though thisobservation is for its directness it is only applicable to the topos situation where X is openCorollary 115 is available more generally

Kockrsquos student and co-author Christian Mikkelsen constructed finite colimits using the min-imal topos axioms However their existence follows much more easily from the next result dueto Robert Pare [Par74] which was the inspiration for the present work (It is reproduced inMikkelsenrsquos thesis [Mik76 p 57]) Nevertheless Mikkelsenrsquos characterisation of X+Y sub ΩXtimesΩY

is much simpler than the one obtained by unwinding the monadic result [Tay99] uses it in Ex-ample 217 before defining categories in Section 41 and monads in Section 75

Theorem 113 For a topos Ω(minus) a Ω(minus) is monadic

Proof Since every object X is discrete ηX is mono by Lemma 612 so Ω(minus) is faithful [Mac71sectIV 3] But as ηX is classified (open) it is regular mono so Ω(minus) also reflects invertibility (and Xis replete in the sense of synthetic domain theory)

We know that the equaliser

X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA

exists in any topos (since ΩΩΩA

is discrete) but we need to show that Ωi is also the coequaliserIn fact there is a split coequaliser diagram with existηΩA

as the other mapSince ΩηA middotηΩA = ΩηA middotΩα = idΩA (it is called coreflexive) the square on the left is a pullback

X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6

-existηΩA- ΩΩΩΩA

ΩΩα

6

As i and ηΩA are mono theyrsquore open and admit existential quantification satisfying the BeckndashChevalley condition on the right This says that Ωi and existi split the idempotent ΩΩα middot existηΩA

since

also Ωi middot existi = id This idempotent is the one that arises from the split coequaliser diagram sinceΩηΩA middot existηΩA

= id and Ωα middot i = ηΩA middot i Hence by Beckrsquos theorem the adjunction is monadic

Remark 114 In the monadic situation

X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX

5Actually a description in the sense of Russell cf [Tay99 sect12] See also Section A 9

43

is always an Σ-split equaliser What is peculiar to the topos case is that the quantifiers exist andare splittings In fact the universal quantifier could be used instead and these are the least andgreatest splittings

existηX 6 ηΩX 6 forallηX existηΩΩX

6 ΩΩηΩX 6 forallηΩΩX

The dual of the Euclidean principle is not needed to use the universal quantifier since the argumentis based on the BeckndashChevalley condition rather than the Frobenius law cf Proposition 311

Corollary 115 Any topos E is cartesian closed

Proof Since it is a right adjoint (minus)Y

preserves equalisers More precisely we first constructthe equaliser

XY-ηYX - Σ(ΣX times Y )

Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)

(since all finite limits exist in E) and then a little easy diagram chasing shows that it also has therequired universal property of the exponential

Conjecture 116 If all ηX are open inclusions then E is a topos (If existη1exists then negation is

definable if forallη1 exists then implication appears to be definable)

Now we shall look for a converse to Parersquos theorem ie given a monadic category what furtherconditions would force it to be an elementary topos These conditions will be in the form of theexistence of quantifiers in higher order logic

Lemma 117 The object Ω is discrete iff it is an internal Heyting algebra

Proof Define x =Ω y as usual by (xrArr y)and (y rArr x) and conversely xrArr y by((xand y) =Ω x

)

Although it follows that all powers of Ω are Heyting algebras (hArr) ΩX times ΩX rarr ΩX has thewrong type to be the equality predicate

Proposition 118 Every object of a topos is compact ie it has a universal quantifier [We donot mean the sense of Dubuc and Penon [DP86] which also requires the dual Frobenius law]

Proof By Proposition 710 since the singleton λxgt sub ΩX is classified

Lemma 119 If Ω is discrete and ΩX is compact then ΩΩX is also discrete

Proof Leibnizian equality F =ΩΩX G iff forallφΩX Fφ =Ω Gφ

Lemma 1110 If X is T0 (ie ηX X rarr ΩΩX is mono) and ΩΩX is discrete then X is alsodiscrete

Proof Proposition 611

There is another famous reduction amongst quantifiers in higher order logic

Lemma 1111 If Ω is discrete and both Ω and X are compact then X is overt

Proof existx φ[x] is forallσ(forallx (φ[x]rArr σ)

)rArr σ

44

Theorem 1112 Let Ω be a Euclidean semilattice in a category E such that the adjunctionΩ(minus) a Ω(minus) is monadic Then the following are equivalent(a) E is a topos with subobject classifier Ω

(b) all objects are overt and discrete

(c) all objects are overt Ω is discrete and all ΩX are compact

(d) Ω is discrete and all objects are compactHence E is also a pretopos and cartesian closed

Proof [arArrd] Proposition 118 [drArrc] Lemma 1111 [crArrb] Lemma 1110[brArra] Condition (b) says that the pretopos of overt discrete objects that we discussed in the

previous section is in fact the whole of E Hence all maps are open (Lemma 102) and in particularΩ classifies all monos by Theorem 310 and Corollary 103 so E is a topos

Remark 1113 From this point of view CDLatop falls short of being a topos in that Υ is notdiscrete but the discrete objects are sets (Examples 212 and 614)

Putting this Theorem together with Theorem 42 and Remark 43 we have justified the claimthat the monadicity property plays the role of comprehension in the sense that it provides anew formalism for elementary toposes that makes no mention of subsets and doesnrsquot even needdependent types

Corollary 1114 The free category with an exponentiating Heyting algebra such that the ad-junction is monadic all objects are overt and all algebras are compact is a topos [This suffersfrom the same error as Theorem 42]

Remark 1115 In terms of our common formulation of geometric and higher order logic we haveseen that the difference between them is measured by the availability of quantifiers of various

kinds I feel that some of these quantifiers (in particular forallΩX ΩΩX rarr Ω from which we deduced

discreteness of ΩΩX ) take the atomic theory of matter beyond what is justified by our intuitionof ldquocollectionsrdquo and other mathematical objects (such as Abelian groups with bases of formaltriangulations from which homology and category theory developed)

Whilst a great deal of the geometrical core of mathematics could potentially be developedwithin our geometric logic there are some things that cannot be done in this logically weakscheme notably questions of well-foundedness termination strong normalisation and consistencyin recursion theory and proof theory We might try to formulate an intermediate scheme ofquantifiers to handle these matters retaining the Euclidean and monadic conditions as the basicframework

At first sight the above results would appear to rule this out if rArr is to be allowed and forallΩX

forbidden but something similar to the latter is to be included However synthetic domain theoryhas already shown that a model may have two objects Σ and Ω classifying weaker and strongerfragments of logic (Remark 213) The extra quantifier could make use of both objects but sinceall maps Ωrarr Σ are constant we are left with

forall ΠΣXminusrarr Π

where Π is the name of the intermediate classifier with Σ sub Π sub Ω We might hope to use itto give a synthetic proof of the general recursion theorem that the induction scheme suffices forrecursion [Tay99 sect63]

But what is already clear from these investigations is that our ldquosyntheticrdquo arguments in topol-ogy are much simpler and to the point than the traditional ones in point-set topology locale theory

45

and continuous lattices So long as we give up trying to detect equality of predicates they alsohave a programming interpretation

Acknowledgements The original inspiration for this work came while I was visiting PinoRosolini at the Universita degli Studi di Genova in March 1993 and many of the results werefound during my second visit in October and November 1997 both visits were funded by theItalian Concilio Nazionale della Ricerca

This paper was drafted during my visit to Macquarie University in August 1997The work has benefitted much from my discussions with Martın Escardo Martin Hyland

Peter Johnstone Anders Kock Eugenio Moggi Bob Pare Bob Rosebrugh Pino Rosolini AndreaSchalk Alex Simpson Thomas Streicher Steve Vickers Graham White Todd Wilson and RichardWood

The Euclidean principle was announced in the categories electronic mail forum on 29 June1997 but I had first encountered this equation on 12 March 1997

References

[Bar88] Robert Barnhart Chambers Dictionary of Etymology 1988

[Bar92] Henk Barendregt Lambda calculi with types In Samson Abramsky et al editorsHandbook of Logic in Computer Science volume 2 pages 117ndash309 Oxford UniversityPress 1992

[Ber78] Gerard Berry Stable models of typed λ-calculi In International Colloquium onAutomata Languages and Programming Lecture Notes in Computer Science pages72ndash89 Springer-Verlag 1978

[BFS03] Anna Bucalo Carsten Fuhrmann and Alex Simpson An equational notion of liftingmonad Theoretical Computer Science 29431ndash60 2003

[Bou66] Nicolas Bourbaki Topologie Generale Hermann 1966 English translation ldquoGeneralTopologyrdquo distrubuted by Springer-Verlag (1989)

[BP80] Michael Barr and Robert Pare Molecular toposes Journal of Pure and AppliedAlgebra 17127ndash132 1980

[BR98] Anna Bucalo and Giuseppe Rosolini Repleteness and the associated sheaf Journal ofPure and Applied Algebra 127147ndash151 1998

[BW85] Michael Barr and Charles Wells Toposes Triples and Theories Springer-VerlagBerlin Germany Heidelberg Germany London UK etc 1985

[CLW93] Aurelio Carboni Steve Lack and Robert Walters Introduction to extensive anddistributive categories Journal of Pure and Applied Algebra 84145ndash158 1993

[Coc93] J Robin B Cockett Introduction to distributive categories Mathematical Structuresin Computer Science 3277ndash307 1993

[CW87] Aurelio Carboni and Robert Walters Cartesian bicategories I Journal of Pure andApplied Algebra 49(1ndash2)11ndash32 1987

[DP86] Eduardo Dubuc and Jacques Penon Objets compacts dans les topos Journal of theAustralian Mathematical Society Series A 40203ndash217 1986

[Esc99] Martın Escardo On the compact regular coreflection of a stable compact locale InProceedings of the 15th conference on Mathematical Foundations of Programming

46

Semantics (MFPS XV) volume 20 of Electronic Notes in Theoretical ComputerScience Elsevier 1999

[Esc04] Martın Escardo Synthetic topology of data types and classical spaces ElectronicNotes in Theoretical Computer Science 8721ndash156 2004

[FR97] Marcelo Fiore and Giuseppe Rosolini Two models of synthetic domain theoryJournal of Pure and Applied Algebra 116151ndash162 1997

[Fre66a] Peter Freyd Algebra-valued functors in general and tensor products in particularColloq Math 1489ndash106 1966

[Fre66b] Peter J Freyd The theory of functors and models In Theory of Models mdashProceedings of the 1963 International Symposium at Berkeley pages 107ndash120 NorthHolland 1966

[FS90] Peter Freyd and Andre Scedrov Categories Allegories volume 39 of North-HollandMathematical Library North-Holland Amsterdam 1990

[FW90] Barry Fawcett and Richard Wood Completely distributive lattices I MathematicalProceedings of the Cambridge Philosophical Society 10781ndash9 1990

[GHK+80] Gerhard Gierz Karl Heinrich Hofmann Klaus Keimel Jimmie D Lawson MichaelMislove and Dana S Scott A Compendium of Continuous Lattices Springer-Verlag1980 Second edition Continuous Lattices and Domains published by CambridgeUniversity Press 2003

[Hau14] Felix Hausdorff Grundzuge der Mengenlehre 1914 Chapters 7ndash9 of the first editioncontain the material on topology which was removed from later editions Reprintedby Chelsea 1949 and 1965 there is apparently no English translation

[Hyl91] J Martin E Hyland First steps in synthetic domain theory In Aurelio CarboniMaria-Cristina Pedicchio and Giuseppe Rosolini editors Proceedings of the 1990Como Category Conference number 1488 in Lecture Notes in Mathematics pages131ndash156 Springer-Verlag 1991

[Hyl10] Martin Hyland Some reasons for generalising domain theory MathematicalStructures in Computer Science 20(02)239ndash265 2010

[JM07] Achim Jung and Andrew Moshier A Hofmann-=Mislove theorem for bitopologicalspaces Electronic Notes in Theoretical Computer Science 173159ndash175 2007

[Joh77] Peter T Johnstone Topos Theory Number 10 in London Mathematical SocietyMonographs Academic Press 1977

[Joh82] Peter T Johnstone Stone Spaces Number 3 in Cambridge Studies in AdvancedMathematics Cambridge University Press 1982

[JT84] Andre Joyal and Myles Tierney An extension of the Galois theory of GrothendieckMemoirs of the American Mathematical Society 51(309) 1984

[KM74] Anders Kock and Christian Mikkelsen Non-standard extensions in the theory oftoposes In Albert Hurd and Peter Loeb editors Victoria Symposium onNonstandard Analysis number 369 in Lecture Notes in Mathematics pages 122ndash143Springer-Verlag 1974

[Koc81] Anders Kock Synthetic Differential Geometry Number 51 in London MathematicalSociety Lecture Notes Cambridge University Press 1981 Second edition number333 2006

[Law71] Bill Lawvere Quantifiers and sheaves In Actes du Congres International desMathematiciens volume 1 pages 329ndash334 Gauthier-Villars 1971

[Law00] Bill Lawvere Comments on the development of topos theory In Jean-Paul Piereditor Development of Mathematics 1950ndash2000 Birkhauser 2000

47

[LR75] Joachim Lambek and Basil Rattray Localizations and sheaf reflectors Transactionsof the American Mathematical Society 210275ndash293 1975

[Mac71] Saunders Mac Lane Categories for the Working Mathematician Springer-VerlagBerlin 1971

[Mar47] Andrei Markov On the representation of recursive functions Doklady Akademii NaukSSSR 581891ndash2 1947

[Mik76] Christian Mikkelsen Lattice-theoretic and Logical Aspects of Elementary Topoi PhDthesis Arhus Universitet 1976 Various publications number 25

[ML84] Per Martin-Lof Intuitionistic Type Theory Bibliopolis Naples 1984

[MR77] Michael Makkai and Gonzalo Reyes First Order Categorical Logic Model-TheoreticalMethods in the Theory of Topoi and Related Categories Number 611 in Lecture Notesin Mathematics Springer-Verlag 1977

[MRW02] Francisco Marmolejo Robert Rosebrugh and Richard Wood A basic distributivelaw Journal of Pure and Applied Algebra 168209ndash226 2002

[Par74] Robert Pare Colimits in topoi Bulletin of the American Mathematical Society80(3)556ndash561 May 1974

[Pho90a] Wesley Phoa Domain Theory in Realizability Toposes PhD thesis University ofCambridge 1990 University of Edinburgh Dept of Computer Science reportCST-82-91 and ECS-LFCS-91-171

[Pho90b] Wesley Phoa Effective domains and intrinsic structure In Logic in Computer Science5 pages 366ndash377 IEEE Computer Society Press 1990

[Ric56] Henry Rice Recursive and recursively enumerable orders Transactions of theAmerican Mathematical Society 83277 1956

[Ros86] Giuseppe Rosolini Continuity and Effectiveness in Topoi D phil thesis Universityof Oxford 1986

[RT98] Giuseppe Rosolini and Paul Taylor Abstract stone duality in synthetic domaintheory In preparation 1998

[Sam87] Giovanni Sambin Intuitionistic formal spaces a first communication In D Skordeveditor Mathematical Logic and its Applications pages 187ndash204 Plenum 1987

[Tay87] Paul Taylor Homomorphisms bilimits and saturated domains mdash some very basicdomain theory 1987

[Tay90] Paul Taylor An algebraic approach to stable domains Journal of Pure and AppliedAlgebra 64171ndash203 1990

[Tay91] Paul Taylor The fixed point property in synthetic domain theory In Gilles Kahneditor Logic in Computer Science 6 pages 152ndash160 IEEE 1991

[Tay99] Paul Taylor Practical Foundations of Mathematics Number 59 in Cambridge Studiesin Advanced Mathematics Cambridge University Press 1999

[Ver94] Japie J C Vermeulen Proper maps of locales Journal of Pure and Applied Algebra9279ndash107 1994

[Vic98] Steven Vickers Topical categories of domains Mathematical Structures in ComputerScience 1998

[Wil94] Todd Wilson The Assembly Tower and some Categorical and Algebraic Aspects ofFrame Theory PhD thesis CarnegiendashMellon University May 1994

[Woo04] Richard J Wood Ordered sets via adjunctions In Maria-Cristina Pedicchio andWalter Tholen editors Categorical Foundations number 97 in Encyclopedia ofMathematics and its Applications pages 1ndash47 Cambridge University Press 2004

48

The papers on abstract Stone duality may be obtained fromwwwPaul TaylorEUASD

[O] Paul Taylor Foundations for Computable Topology in Giovanni Sommaruga (ed)Foundational Theories of Mathematics Kluwer 2011

[A] Paul Taylor Sober spaces and continuations Theory and Applications of Categories10(12)248ndash299 2002

[B] Paul Taylor Subspaces in abstract Stone duality Theory and Applications of Categories10(13)300ndash366 2002

[D] Paul Taylor Non-Artin gluing in recursion theory and lifting in abstract Stone duality 2000

[E] Paul Taylor Inside every model of Abstract Stone Duality lies an Arithmetic UniverseElectronic Notes in Theoretical Computer Science 122 (2005) 247-296

[F] Paul Taylor Scott domains in abstract Stone duality March 2002

[Gndash] Paul Taylor Local compactness and the Baire category theorem in abstract Stone dualityElectronic Notes in Theoretical Computer Science 69 Elsevier 2003

[G] Paul Taylor Computably based locally compact spaces Logical Methods in ComputerScience 2 (2006) 1ndash70

[Hndash] Paul Taylor An elementary theory of the category of locally compact locales APPSEMWorkshop Nottingham March 2003

[H] Paul Taylor An elementary theory of various categories of spaces and locales November2004

[I] Andrej Bauer and Paul Taylor The Dedekind reals in abstract Stone duality MathematicalStructures in Computer Science 19 (2009) 757ndash838

[J] Paul Taylor A λ-calculus for real analysis Journal of Logic and Analysis 2(5) 1ndash115 (2010)

[K] Paul Taylor Interval analysis without intervals February 2006

[L] Paul Taylor Tychonovrsquos theorem in abstract Stone duality September 2004

[N] Paul Taylor Computability in locally compact spaces 2010

[AA] Paul Taylor Equideductive categories and their logic 2010

[BB] Paul Taylor An existential quantifier for topology 2010

[CC] Paul Taylor Cartesian closed categories with subspaces 2009

[DD] Paul Taylor The Phoa principle in equideductive topology 2010

[EE] Paul Taylor Discrete mathematics in equideductive topology 2010

[FF] Paul Taylor Equideductive topology 2010

[GG] Paul Taylor Underlying sets in equideductive topology 2010

49

Introduction

Support classifiers

The Euclidean principle

The origins of the Euclidean principle

The Phoa Principle

Discrete and Hausdorff objects

Overt and compact objects

The quantifiers

Unions and coproducts

Open discrete equivalence relations

Monadicity for elementary toposes

treated in the same way as the collections of open and recursively enumerable subsets in topologyand recursion theory The underlying formulation in which we do this is a category with a ldquotruthvaluesrdquo object Σ for which the adjunction Σ(minus) a Σ(minus) is monadic Robert Pare proved in 1974that any elementary topos has this property whilst the category LKLoc of locally compact localeshas it too (sectO 510 and Theorem B 3111) [B] explains how this is an abstraction of Stone duality

Gerhard Gentzenrsquos natural deduction was the first principled treatment of the logical connec-tives and Per Martin-Lof used the CurryndashHoward isomorphism to extend it in a principled way totype constructors [ML84] However the natural conclusion of such an approach is to say that theexistential quantifier is the same as the dependent sum ie that a proof of existx φ(x) must alwaysprovide a witness a particular a for which φ(a) holds

This conflicts with geometrical usage in which we may say that the Mobius band has two edgesor a complex number two square roots locally but not globally ie there exists an isomorphismbetween 2 equiv 1 + 1 and the set of edges or roots on some open subspace [Tay99 sect24] Similarlyinterprovable propositions are for Martin-Lofrsquos followers isomorphic types not equal ones andtheir account of the powerset is a bureaucratic one a structure within which to record the historiesof formation and proof of the propositionndashtypes [op cit sect95]

The way in which category theory defines the powerset is not perhaps based so firmly on alogical creed as is Martin-Lof type theory in that it describes provability rather than proof but itwas at least designed for the intuitions of geometry and symmetry This notion mdash the subobjectclassifier in an elementary topos which is readily generalised to the classifier Σ for open subsets(the Sierpinski space) and recursively enumerable ones mdash then obeys the curious equation

σ and F (σ)hArr σ and F (gt) for all σ isin Σ and F Σrarr Σ

which we call the Euclidean principle The Frobenius law which is part of the categoricalformulation of the (geometrical) existential quantifier and was so called by Bill Lawvere is anautomatic corollary of the Euclidean principle From this we develop the connectives of first ordercategorical logic in particular stable effective quotients of equivalence relations

Whilst set theory and topology have common historical roots [Hau14] the motivation for acommon treatment of the kind that we envisage is Marshall Stonersquos dictum that we should ldquoalwaystopologizerdquo mathematical objects even though they may have been introduced entirely in termsof discrete ideas [Joh82 Introduction] For example the automorphisms of the algebraic closureof Q form not an infinite discrete (Galois) group but a compact topological group Similarly thepowerset of even a discrete set is not itself a discrete set but a non-Hausdorff topological lattice(Steven Vickers has taken the same motivation in a different direction [Vic98])

The types in our logic are therefore to be spaces The topological structure is an indissolublepart of what it is to be a space it is not a set of points together with a topology any more thanchipboard (which is made of sawdust and glue) is wood

When we bring (not necessarily Martin-Lof) type theory together with categorical logic [Tay99]logical notions such as the quantifiers acquire meanings in categories other than Set In par-ticular with the internal lattice ΣX in place of the powerset of X the internal adjunctionsexistX a Σ a forallX where they exist suggest interpretations of the quantifiers We shall find thatthey obey the usual logical rules but in the topological setting they also say that the space X isrespectively overt (a word that we propose to replace one of the meanings of open) or compact

It is well known that the recursively enumerable subsets of N almost form a topology sincewe may form finite intersections and certain infinitary unions of ldquoopenrdquo subsets However theunification of topology with recursion theory ie making precise Dana Scottrsquos thesis that continuityapproximates the notion of computability involves a revolutionary change because the classical

1The letters denote the other papers in the ASD programme

2






Cop

C

Σ(minus)

6a Σ(minus)






3












4





V - U

Y

flowasti

cap f - X

i

cap




U - 1

X

i

cap

φ- Σ

gt



U subi - X

gt -

φ- Σ





5












6


V - U - 1

Y

cap f - X

cap

φ - Σ

gt





tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-


1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7
















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers









Cop

C

Σ(minus)

6a Σ(minus)






3












4





V - U

Y

flowasti

cap f - X

i

cap




U - 1

X

i

cap

φ- Σ

gt



U subi - X

gt -

φ- Σ





5












6


V - U - 1

Y

cap f - X

cap

φ - Σ

gt





tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-


1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7
















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers















4





V - U

Y

flowasti

cap f - X

i

cap




U - 1

X

i

cap

φ- Σ

gt



U subi - X

gt -

φ- Σ





5












6


V - U - 1

Y

cap f - X

cap

φ - Σ

gt





tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-


1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7
















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers








V - U

Y

flowasti

cap f - X

i

cap




U - 1

X

i

cap

φ- Σ

gt



U subi - X

gt -

φ- Σ





5












6


V - U - 1

Y

cap f - X

cap

φ - Σ

gt





tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-


1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7
















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers















6


V - U - 1

Y

cap f - X

cap

φ - Σ

gt





tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-


1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7
















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers





V - U - 1

Y

cap f - X

cap

φ - Σ

gt





tw- Σtimes Σ

Σtimes Σ

and times id

and - Σ

and

Σ

and

and -

Σtimes Σ Σ

Σid-

(gt id)

-

Σ

and id

Σ

(idgt)

Σtimes Σand -

∆

Σ

id

-


1 - 1

1 Σ

gt

cap

Σ

gt π1

Σtimes Σ

idtimesgt

cap

and - Σ

gt

U cap V sub - U 1

1 -

-

-

cap

Σ

-

V

cap

sub

-

- X

ψ -

φ and ψ- Σ

Σ

-

φ

-

Σtimes Σ and

--

7
















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers



















8












φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers















φ(x) and F(x φ(x)


9



[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






[φ]

cap

sub - X

cap

(idgt)-

(id φ)- X times Σ

F - Σ

gt

cap

sim=

- [φ] cap [Fφ]cup

6

sub - [Fφ]cup

6

- 1

gtcup

6




W - V - 1 ============= 1

U times ΣX

cap


cap

evU - Σ

gt

X times ΣX

itimes id

cap


itimes id

cap

I - Σ

gt

id

-




10



U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






U equiv [φ] subi

- Xsub



ΣU

Σigt-I- ΣX






(x φ(x)


Euclidean principle




E(λx F(λψ ψx)



= E(λx F(λψ ψx)






11












)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers















)= I middot P

(I(θ) and ψ















12


V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers





V - 1

1 U

cap

ψ - Σ

gt

cap

Σ

gt

cap

φX

cap

I(ψ)

-




Proposition 28) iff



V - 1

U

sim=

cap

ψ - Σ

gt


C(UΣ)-


C(XΣ) darr φ-

(minus) and φ

C(XΣ)

Sub(U)

pullback


pullback

-

(minus) cap USub(X)

pullback

13









Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers












Vg - U - 1 ΣV

ΣgΣU3 θ

Y

j

cap

f - X

i

cap

φ - Σ

gt

ΣY

existj

ΣfΣX

existi




X ΣX

Usub

-

V

sup

ΣU-

-

ΣV

--

U cap Vsub

-

sup

ΣUcapV-

---


14








hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers











hcf(n (f + n)

)= hcf

(n (f + 0)

)

In fact hcf(F (n) n

)= hcf

(F (0) n








15








(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers











(e) G(H(σ) σ















)hArr rec


)








16




or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers







or or or or











17


Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers





Fminus1(gt)














or is still running

18





perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers








perpΓ


UI equiv Γ + U

[id i] - Γ

6 V -

-

1-

Γ + Γ

id + i

cap

[perpgt] - Σ

σ

τ

Γ

τ -

i

-

Σ

gt










19

















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers




















(Fperpor (σandFgt)




20
























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers



























21





X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers








X - 1

X timesX

∆

cap

(=X)- Σ

gt












22
















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers



















23



Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






Γtimes [x =X y]

cap


cap

p01-

p02

- ΓtimesX

cap

φ - Σ

gt


6


6

sub - [φy]cup

6

- 1

gt

6



















24


U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers





U - X - 1

U times U

- X timesX

cap

- Σ




Γb -

a- X

X - ΣX

ΣX timesX

Γ〈b a〉 -

〈a a〉- X timesX

=X -

X times idX-

Σ

ev

X

∆

cup

6

-

a

-

1

gt

6



X X - ΣX

ΣΣX

ηX

ΣΣ X

- ΣΣΣX

ηΣX

ΣηX

66



25



Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






Set

components

perp- discrete -

perpunderlying set

Pos

Ω Υ





Set

(b)

perp- -perp(a)

LKLoc

(d) -perp perp(e)

KHaus

Ω (ac)

Σ(ef)

- 2










26























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers


























27
















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers



















28





1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers








1 - 1 1 - 1

ΣX

perp

existX - Σ

perp

ΣX

gt

forallX - Σ

gt


existX



U

V -

-

1 --

1

Γ

iV

ψ -orφ-

iU

-

ΣX

gt

forallX - Σ

gt


1 - 1

Σ

gt

forallX middot Σ-

id- Σ

gt

29




X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers







X - X - 1 (φ x) | φ[x] - 1

sub

ΣX timesX

(gt id)

cap


(gt id)

cap

π0- Σ

gt

ΣX timesX

cap

ev- Σ

gt











X- X - ΣX




30

8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers




8 The quantifiers






ΣXtimesY Σp0

ΣX

Z

p1

f - Y

p1

- 1

ΣZ

G

Σp1

6

ΣfΣY

F

Σp1

6

Σ

Σ

E

a Σ

6


(λx ω(x y)


F(Σp1(ψ)



)hArr F (Σp1ω)(y)


(λx ω(x fz)







(φ and Σ(σ)

)

Γtimes ΣF - Σ

ΣX times ΣX

φtimes Σ

andX - ΣX

E

6



31



(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






(λx φ(x z)



)= λz












X

p0

f - Y

p0

A equiv ΣX

orI

Σf

B equiv ΣY

orI








32






Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers









Γ ` φ prop

Γ x X ` φ prop

Γ φ ` θ

Γ x Xφ ` θ



Γ x X ` φ(x) prop


Γ ` θ prop Γ x Xφ(x) ` θ=========================

Γexistx φ(x) ` θ

















)or(existy ψ(y)

)

33



(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






(existxNinfin ψ(x)

)hArr(






Γ - X - 1

ΓtimesX - 1

λxgt -

--

1

-

ΩX forallΩ

- Ω

gt

Σ

φ

- Ω

gt

ΣX

cap

forallΣ-

φ

-

-

Σ

gt

mon

o-




34












X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers















X - Z Y

1

- 2

s

1



X - Z

1 ν0 - 2

s

Σ

φ

-

(gtperp

)-gt -

35





C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers








C







f1 + f2 X1 +X2 rarr Y1 + Y2

Proof We define





X + Y - 1


cap


gt



36










P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers













P - X

Y

glowastf

g - D

f

Proof

D - 1

P - X

f -

D timesD

cap

=D- Σ

gt

X times Y

i


(id f)

cap

f times id -

X-

glowastf

- Y

p1

g - D

p1

1

-

37





Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers








Xid - X ΣX

id - ΣX

X

id

i - D

i

ΣX

id

6

existi - ΣD

Σi

6













38


∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers





∆

Q q

X p0

p1

K

X timesX

i

p1

p0








and reflexivity

U- Σq -

existqΣX

Σp0

-- Σp1 -

existp1

ΣK

ΣXtimesX

Σi

existi--

existp1

-

Σ p1

-

Σp0

-






(δ(y z) and φ(z)






39








K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers











K subi - X timesX

X

p0

p1

QtimesX

q timesX

X timesQ

X times q

--

Q

q

- ∆ - QtimesQ

q timesQ

q timesQ --














40


Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers





Xq -- Q-

Q- ΣQ-Σq- ΣX















)and τ(x z)


)and σ(y z)






41











42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers














42



X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






X - 1 XY - 1

ΩX

X

cap

sX - Ω

gt

ΩXtimesY

(sX)Y- ΩY

gt







X-i - ΩA

Ωα --

ηΩA

- ΩΩΩA




X-i - ΩA ΩX-

existi- ΩΩA

ΩA

i

- ηΩA- ΩΩΩA

Ωα

ΩΩA

Ωi

6


ΩΩα

6


since




X-ηX - ΣΣX

ΣΣηX-

-ηΣΣX

- ΣΣΣΣX


43









Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers












Σ(ΣηX times Y )-

-ηY

ΣΣX

- Σ

(ΣΣΣX

times Y)







)











)rArr σ

44



















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers






















45






References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers









References














46




















47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers























47























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers


























48
























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers



























49

Introduction

Support classifiers



The Phoa Principle



The quantifiers




Geometric and Higher Order Logic in terms of Abstract ...pt/ASD/geohol/geohol.pdf1 Introduction The...

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Transcript of Geometric and Higher Order Logic in terms of Abstract ...pt/ASD/geohol/geohol.pdf1 Introduction The...