Glossary of Set Theory

Glossary of set theoryFrom Wikipedia, the free encyclopediaThis is a glossary of set theory.Contents :Greek !$@ A B C D E F G H I J K L M N O P Q R S T U V W XYZ See also ReferencesGreek

Often used for an ordinal

X is the Stoneech compactification of X

A gamma number, an ordinal of the form

A delta number, an ordinal of the form

(Greek capital delta, not to be confused with a triangle )1. A set of formulas in the Levy hierarchy2. A delta system

An epsilon number, an ordinal with =

1. The order type of the rational numbers2. An eta set, a type of ordered set

The order type of the real numbers

Often used for a cardinal

1. Often used for a cardinal2. The order type of the real numbers

A measure

1. A product of cardinals2. A set of formulas in the Levy hierarchy

The rank of a set

countable, as in -compact, -complete and so on

1. A sum of cardinals2. A set of formulas in the Levy hierarchy

The smallest infinite ordinal

Glossary of set theory - Wikipedia, the free encyclopedia http://en.wikipedia.org/w/index.php?title=Glossary_of_set_theory&pri...

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The largest ordinal, the class of all ordinals that are sets

!$@, =, , , , , , ,

Standard set theory symbols with their usual meanings (is a member of, equals, is contained in, contains, properly contains, isproperly contained in, union, intersection, empty set)

Standard logical symbols with their usual meanings (and, or, implies, is equivalent to, not, for all, there exists)

fX is the restriction of a function f to some set X

(A triangle, not to be confused with the Greek letter )1. The symmetric difference of two sets2. A diagonal intersection

The diamond principle

The square principle

The composition of functions

sx is the extension of a sequence s by x

+1. Addition of ordinals2. Addition of cardinals3. + is the smallest cardinal greater than 4. B+ is the poset of nonzero elements of a Boolean algebra B

The difference of two sets: xy is the set of elements of x not in y.

A product of sets

/A quotient of a set by an equivalence relation

1. xy is the ordinal product of two ordinals2. xy is the cardinal product of two cardinals

*An operation that takes a forcing poset and a name for a forcing poset and produces a new forcing poset.

The class of all ordinals, or at least something larger than all ordinals

1. Cardinal exponentiation2. Ordinal exponentiation

1. The set of functions from to


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1. Implies2. f:XY means f is a function from X to Y.3. The ordinary partition symbol, where ()nm means that for every coloring of the n-element subsets of with m colorsthere is a subset of size all of whose n-element subsets are the same color.

f ' xIf there is a unique y such that x,y is in f then f ' x is y, otherwise it is the empty set. So if f is a function and x is in its domain,then f ' x is f(x).

f Xf X is the image of a set X by f. If f is a function whose domain contains X this is {f(x):xX}

[ ]1. M[G] is the smallest model of ZF containing G and all elements of M.2. [] is the set of all subsets of a set of cardinality , or of an ordered set of order type 3. [x] is the equivalence class of x

{ }1. {a, b, ...} is the set with elements a, b, ...2. {x : (x)} is the set of x such that (x)

a,b is an ordered pair, and similarly for ordered n-tuples

|X|The cardinality of a set X

||||The value of a formula in some Boolean algebra

(Quine quotes, unicode U+231C, U+231D) is the Gdel number of a formula

A means that the formula follows from the theory A

A means that the formula holds in the model A

The forcing relation

An elementary embedding

pq means that p and q are incompatible elements of a partial order

The Hebrew letter aleph, which indexes the aleph numbers or infinite cardinals

The Hebrew letter beth, which indexes the beth numbers

A serif form of the Hebrew letter gimel, representing the gimel function

The Hebrew letter Taw, used by Cantor for the class of all cardinal numbersA


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The least size of a maximal almost disjoint family of infinite subsets of

AThe Suslin operation

ACThe Axiom of choice

ADThe axiom of determinacy

AdmissibleAdmissible set

AHAleph hypothesis, a form of the generalized continuum hypothesis

aleph1. An infinite cardinalThe aleph function taking ordinals to infinite cardinals

almost universalA class is called almost universal if every subset of it is contained in some member of it

antichainAn antichain is a set of pairwise incompatible elements of a poset

Aronszajn treeAn Aronszajn tree is an uncountable tree such that all branches and levels are countable

atom1. An urelement, something that is not a set but allowed to be an element of a set2. An element of a poset such that any rwo elements smaller than it are compatible.

axiomAczel's anti-foundation axiomAD+Axiom of adjunctionAxiom of amalgamation Same as axiom of unionAxiom of choiceAxiom of comprehensionAxiom of constructibilityAxiom of determinacyAxiom of empty setAxiom of extensionality or axiom of extentAxiom of foundation Same as axiom of regularityAxiom of global choiceAxiom of heredity (any member of a set is a set; used in Ackermann's system.)Axiom of infinityAxiom of limitation of sizeAxiom of pairingAxiom of power setAxiom of real determinacyAxiom of regularityAxiom of replacement Same as axiom of substitutionAxiom of subsets Same as axiom of powersetsAxiom of substitutionAxiom of unionAxiom schema of predicative separationAxiom schema of replacementAxiom schema of separationAxiom schema of specification Same as axiom schema of separationFreiling's axiom of symmetryMartin's axiom


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Proper forcing axiomB

The bounding number, the least size of an unbounded family of sequences of natural numbersBerkeley cardinalB

A Boolean algebraBA

Baumgartner's axiom, one of three axioms introduced by Baumgartner.BG

BernaysGdel set theory without the axiom of choiceBGC

BernaysGdel set theory with the axiom of choicebounding number

The bounding number is the least size of an unbounded family of sequences of natural numbersBurali-Forti paradox

The ordinal numbers do not form a setCc

The cardinality of the continuumC

The Cantor setCantor

1. The Cantor normal form of an ordinal is its base expansion.2. Cantor's paradox says roughly that the cardinal numbers do not form a set.

cardinal1. A cardinal number is an ordinal with more elements than any smaller ordinal

category1. A set of first category is the same as a meager set: a set that is the union of a countable number of nowhere-dense sets, anda set of second category is a set that is not of first category.2. A category in the sense of category theory.

ccccountable chain condition

cfThe cofinality of an ordinal

CHThe continuum hypothesis

classA class is a collection of sets

Club1. Club filter2. Club set3. Clubsuit

cofinalA subset of a poset is called cofinal if every element of the poset is at most some element of the subset.


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cofinalityThe cofinality of a poset (especially an ordinal or cardinal) is the smallest cardinality of a cofinal subset

Colcollapsing algebra

A collapsing algebra Col(,) collapses cardinals between and condensation lemma

Gdel's condensation lemma says that an elementary submodel of an element of the construcible hierarchy is also an elementof the constructible hierarchy

constructibleA set is called constructible if it is in the constructible universe.

continuumThe continuum is the real line or its cardinality

CTMCountable transitive model

cumulative hierarchyA cumulative hierarchy is a sequence of sets indexed by ordinals that satisfies certain conditions and whose union is used as amodel of set theory

D

The dominating number of a posetDC

The axiom of dependent choicedefinable

A subset of a set is called definable set if it is the collection of elements satisfying a sentence in some given languagedelta

1. A delta number is an ordinal of the form 2. A delta system, also called a sunflower, is a collection of sets such that any two distinct sets have intersection X for somefixed set X

diagonal intersectionIf is a sequence of subsets of an ordinal , then the diagonal intersection is

diamond principleJensen's diamond principle states that there are sets A for

ErdsRado theoremThe ErdsRado theorem extends Ramsey's theorem to infinite cardinals

ethereal cardinalExtendible cardinalFF

An F is a union of a countable number of closed setsfinite intersection propertyFIP

The finite intersection property says that the intersection of any finite number of elements of a set is non-emptyfirst category

A set of first category is the same as a meager set: one that is the union of a countable number of nowhere-dense sets.first class

An ordinal of the first class is a finite ordinalFodor's lemma

Fodor's lemma states that a regressive function on a regular uncountable cardinal is constant on a stationary subset.forcing

Forcing (set theory) is a method of adjoining a generic filter G of a poset P to a model of set theory M to obtain a new moelM[G]

GG

1. A generic ultrafilter2. A G is a countable intersection of open sets

gamma numberA gamma number is an ordinal of the form

GCHGeneralized continuum hypothesis

generic1. A generic filter of a poset P is a filter that intersects all dense subsets of P that are contained in some model M.2. A generic extension of a model M is a model M[G] for some generic filter G.

global choiceThe axiom of global choice says there is a well ordering of the class of all sets

HH

Abbreviation for "hereditarily"H

The set of sets that are hereditarily of cardinality less than Hartogs number

The Hartogs number of a set X is the least ordinal such that there is no injection from into X.Hausdorff gap

A Hausdorff gap is a gap in the ordered set of growth rates of sequences of integers, or in a similar ordered setHC

The set of hereditarily countable sets


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hereditarilyIf P is a property the a set is hereditarily P if all elements of its transitive closure have property P. Examples: Hereditarilycountable set Hereditarily finite set

HFThe set of hereditarily finite sets

HSThe class of hereditarily symmetric sets

HODThe class of hereditarily ordinal definable sets

Huge cardinalII0, I1, I2, I3

Certain large cardinal axiomsInaccessible cardinalindecomposable ordinal

An indecomposable ordinal is an ordinal which is not the sum of two smaller ordinalsindescribable cardinalineffable cardinalinner model

An inner model is a transitive model of ZF containing all ordinalsJj

An elementary embeddingJ

Levels of the Jensen hierarchyJensen's covering theorem

Jensen's covering theorem states that if 0

Laver functionA Laver function is a function related to supercompact cardinals that takes ordinals to sets

Lvy1. Azriel Lvy2. The Lvy collapse is a way of destroying cardinals3. The Lvy hierarchy classifies formulas in terms of the number of alternations of unbounded quantifiers

limitA limit cardinal is a cardinal, usually assumed to be nonzero, that is not the successor + of another cardinal A limit ordinal is an ordinal, usually assumed to be nonzero, that is not the successor +1 of another ordinal

M

The smallest cardinal at which Martin's axiom failsM

A model of ZF set theoryMA

Martin's axiomMahlo cardinalmeagermeagre

A meager set is one that is the union of a countable number of nowhere-dense sets. Also called a set of first category.Measurable cardinalMilnerRado paradox

The MilnerRado paradox states that every ordinal number less than the successor + of some cardinal number can bewritten as the union of sets X1,X2,... where Xn is of order type at most n for n a positive integer.

MMMartin's maximum

morassA morass is a tree with ordinals associated to the nodes and some further structure, satisfying some rather complicated axioms.

Mostowski collapseThe Mostowski collapse is a transitive class associated to a well founded extensional set-like relation.

mouseA certain kind of structure used in constructing core models; see mouse (set theory)

NN

1. The set of natural numbers2. The Baire space

naive set theory1. Naive set theory usually means set theory developed non-rigorously without axioms2. An introductory book on set theory by Halmos

normal1. A normal function is a continuous strictly increasing function from ordinals to ordinals2. A normal filter or normal measure on an ordinal is a filter or measure closed under diagonal intersections3. The Cantor normal form of an ordinal is its base expansion.

number classThe first number class consists of finite ordinals, and the second number class consists of countable ordinals.


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O-logic

-logic is a form of logic introduced by Hugh WoodinOCA

The open coloring axiomOD

The ordinal definable setsOn

The class of all ordinalsordinal

An ordinal is the order type of a well-ordered set, usually represented by a von Neumann ordinal, a transitive set well orderedby .

otAbbreviation for "order type of"

P

The pseudo-intersection number, the smallest cardinality of a family of infinite subsets of that has the strong finiteintersection property but has no infinite pseudo-intersection.

P1. The powerset function2. A poset

pairing functionA pairing function is a bijection from XX to X for some set X

pantachiepantachy

A pantachy is a maximal chain of a posetparadox

1. Burali-Forti paradox2. Cantor's paradox3. Hilbert's paradox4. Russell's paradox

PCFAbbreviation for "possible cofinalities", used in PCF theory

PDThe axiom of projective determinacy

permutation modelA permutation model of ZFA is constructed using a group

PFAThe proper forcing axiom

poposet

A set with a partial orderpower

An alternative term for cardinalityproper

1. A proper class is a class that is not a set2. A proper forcing is a forcing notion that does not collapse any stationary set


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3. A proper subset of a set X is a subset not equal to X.QQ

The (ordered set of) rational numbersRR

1. R is an alternative name for the level V of the von Neumann hierarchy.2. The real numbers

Ramsey cardinalran

The range of a functionrank

The rank of a set is the smallest ordinal greater than the ranks of its elementsreflecting cardinalreflection principle

A reflection principle states that there is a set similar in some way to the universe of all setsregular

A regular cardinal is one equal to its own cofinality.Reinhardt cardinalrelation

A set or class whose elements are ordered pairsRO

The regular open sets of a topological space or posetRowbottom cardinalSSCH

Singular cardinal hypothesisScott's trick

Scott's trick is a way of coding proper equivalence classes by sets by taking the elements of the class of smallest ranksecond category

A set of second category is a set that is not of first category: in other words a set that is not the union of a countable number ofnowhere-dense sets.

second classAn ordinal of the second class is a countable infinite ordinal

separating set1. A separating set is a set containing a given set and disjoint from another given set2. A separating set is a set S of functions on a set such that for any two distinct points there is a function in S with differentvalues on them.

separativeA separative poset is one that can be densely embedded into the poset of nonzero elements of a Boolean algebra.

SFIPStrong finite intersection property

SHSuslin's hypothesis


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Shelah cardinalshrewd cardinalsingular

1. A singular cardinal is one that is not regular2. The singular cardinal hypothesis states that if is any singular strong limit cardinal, then 2 = +.

SOCASemi open coloring axiom

Solovay modelThe Solovay model is a model of ZF in which every set of reals is measurable

stationary setA stationary set is a subset of an ordinal intersecting every club set

strong1. The strong finite intersection property says that the intersection of any finite number of elements of a set is infinite2.

Strongly compact cardinalsubtle cardinalsuccessor

1. A successor cardinal is the smallest cardinal larger than some given cardinal1. A successor ordinal is the smallest ordinal larger than some given ordinal

sunflowerA sunflower, also called a delta system, is a collection of sets such that any two distinct sets have intersection X for some fixedset X

SouslinSuslin

Mikhail Yakovlevich SuslinSuslin algebraSuslin cardinalSuslin hypothesisSuslin lineSuslin number. Suslin operationSuslin problemSuslin propertySuslin representationSuslin schemeSuslin setSuslin spaceSuslin theorems about analytic setsSuslin tree

super transitivesupertransitive

A supertransitive set is a set that contains all subsets of all its elementssymmetric model

A symmetric model is a model of ZF (without the axiom of choice) constructed using a group action on a forcing posetTT

A treetall cardinal


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Tarski's theorem1. Tarski's theorem that the axiom of choice is equivalent to the existence of a bijection from X to XX for all sets X

TCThe transitive closure of a set

total orderA total order is a relation that is transitive and antisymmetric such that any two elements are comparable

transitive1. A transitive relation2. The transitive closure of a set is the smallest transitive set containing it.3. A transitive set or class is a set or class such that the membership relation is transitive on it.4. A transitive model is a model of set theory that is transitive and has the usual membership relation

tree1. A tree is a partially ordered set (T,

Zermelo set theory without the axiom of choiceZC

Zermelo set theory with the axiom of choiceZF

ZermeloFraenkel set theory without the axiom of choiceZFA

ZermeloFraenkel set theory with atomsZFC

ZermeloFraenkel set theory with the axiom of choiceSee also

Glossary of Principia MathematicaReferences

Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002 (https://zbmath.org/?format=complete&q=an:1007.03002).

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Glossary of Set Theory

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