On –presentability of some structures of analysis over ...€¦ · On {presentability of some...

On Σ–presentability of some structures of analysisover hereditarily finite superstructures

A.S. Morozov

Sobolev Institute of Mathematicsand

Novosibirsk State UniversityNovosibirsk, Russia

[email protected]

http://www.math.nsc.ru/∼asm256

New Zealand, 6 January 2017 On Σ–presentability

Contents

1 Nonexistence of simple Σ–presentations in hereditarily finitesuperstructures over existentially Steinitz structures for

some automorphism groups and semigroupssome structures of nonstandard analysisInfinite dimensional separable Hilbert spaces

2 A new sufficient condition for the nonexistence of suchpresentations

3 Some open problems


HF(M): a “programmer’s” universe over basic data typeM

HF0(M) = M

HFn+1(M) = HFn(M) ∪ S<ω(HFn(M))

HF(M) =⋃n<ω

HFn(M)

HF(M): all sets which could be explicitly defined using {, },∅, and elements of M, for instance, if M = R:∅, {∅}, {1,

√2}, {r ,∅, {u, π, {∅, {7}}}, 17

√29},

r , u ∈ R, all finite ordinals, etc. . . .


HF(M) as an algebraic structure

Specific membership relation ‘∈’ (elements of M are urelements):

x ∈ y ⇔ (x ∈ HF(M))∧(y ∈ HF(M) \M)∧(x is a member of y)

• U(x) ⇔ x ∈ M

• Language for HF(M):

〈U,∈,∅, 〈Predicates of M〉〉


Σ–definability vs classical computability

Σ–formulas: a special class of formulas that defines ‘computablyenumerable sets’ in HF(M)

computably enumerable ∼ definable by Σ–formulas withparameters (Σ–definable)

computable ∼ ∆–definable ( = Σ– and ¬Σ–definable)


‘Computable model theory over admissible sets’:

Computable structures → Σ–presentable structures

Definition (Yu. L. Ershov)

If each element of a structure A is associated with a set of itscodes (notations) from an admissible set A so that the diagram ofthis structure is a Σ–definable subset of A, then such a coding iscalled a Σ–presentation of A.

If each element of A is coded by a single element of A then such apresentation is called simple.


Definition

An element a ∈M is called ∃–algebraic over A ⊆M if there existsan ∃–formula ϕ(x , y) and parameters b ∈ A such that the setϕM[x , b] is finite and contains a.

CM∃ (A): the set of all algebraic elements over A.

Definition

A structure M is called ∃–Steinitz if CM∃ (A) has the following

exchange property:a ∈ CM

∃ (A ∪ {b}) \ CM∃ (A), implies b ∈ CM

∃ (A ∪ {a}).

Examples: R, C, any model of strongly minimal model completetheory, any model complete field or ordered field, algebraicallyclosed field, etc.


Definition

A set X ⊆M is called independent if for all x ∈ X holdsx /∈ CM

∃ (X \ {x}).

Definition

The dimension of a set: the cardinality of any its maximalindependent set.


Theorem (M., 2014)

Assume that M is an ∃–Steinitz structure of a finite signature. LetA be an arbitrary structure of a finite signature for which thereexist a family (Fi )i<ω of unary operations definable by terms withparameters and a family (Ai )i<ω of subsets of A such that:

1 all the sets Fi [Ai ] are uncountable

2 for any sequence (ai )i<ω ∈∏

i<ω Ai there is an element b ∈ Asuch that for all i < ω holds Fi (b) = Fi (ai ).

Then A is cannot be embedded into a structure having a simpleΣ–presentation over HF(M) with parameters.

Proof: having assumed that such a presentation exists we arrive ata contradiction like ‘a finite set has infinite dimension’.


Theorem (M., 2014)

The following structures are not embeddable into structures havingsimple Σ–presentations with parameters over HF(M) where M isan ∃–Steinitz structure of a finite signature:

1 P(ω), P(ω)/Fin2 The lattice of all open (closed) subsets of Rn, n > 0

3 Sym (ω), Sym (ω)/Fin4 The group (semigroup) of all permutations (mappings)

Σ–definable with parameters over HF(R)

5 The semigroup ωω


Theorem

The following structures are not embeddable into structures havingsimple Σ–presentations with parameters over HF(M) where M isan ∃–Steinitz structure of a finite signature:

1 Aut 〈Q, <〉: the automorphism group of the ordering on therational numbers in the signature with a single operation ofcomposition.

2 Aut 〈R, <〉: the automorphism group of the ordering on thereals in the signature with a single operation of composition.

3 C(Rn): the semigroup of all continuous mappings from Rn toRn, for any n > 0.

4 C1(Rn): the semigroup of all continuously differentiablefunctions from Rn Rn, for any n > 0.


Theorem (A new sufficient condition for nonpresentability)

Assume that M is an ∃–Steinitz structure of a finite signature. LetA be an arbitrary structure of a finite signature such that thereexist a family (Fi )i<ω of unary partial functions definable byΣ–formulas with parameters over HF(A) and a family (Ai )i<ω

of subsets of A such that:

1 for each i < ω holds Ai ⊆ dom (Fi ), and Fi [Ai ] is uncountable

2 for any sequence (ai )i<ω ∈∏

i<ω Ai , there exists a b ∈ A suchthat for all i < ω holds Fi (b) = Fi (ai ).

Then A has no simple Σ–presentations over HF(M) withparameters.


Theorem

Let D be an arbitrary nonprincipal filter over ω and M be an∃–Steinitz structure of a finite signature. Then the filtered power〈Rω/D, St , Inf 〉 extended with unary predicates St (for standardelements, i.e., defined by constant functions from) and Inf (forinfinitesimal elements, i.e., elements situated between all positiveand negative standard elements in Rω/D) has no simpleΣ–presentations over HF(M).

Remark The structures of kind Rω/D are not always real closedfields. For instance, if D is the Frechet filter then this structure isnot linearly ordered.


Nonpresentability of some nonstandard extensions of R(independently of concrete constructions)

Theorem

Let M be an ∃–Steinitz structure of a finite signature and ∗R bean arbitrary extension of the ordered field of the reals R containinginfinitesimal elements; and assume that for each functionf : R2 → R there exists a function ∗f : (∗R)2 → ∗R such that〈R, f 〉f ∈F is an elementary submodel in 〈∗R, ∗f 〉f ∈F , where F isthe family of all binary functions on R. Then the structure〈∗R, St , Inf 〉 in which the predicate St distinguishes standardelements from ∗R (i.e., elements from R) and Inf distinguishesinfinitesimal elements (i.e., Inf =

⋂i<ω

[− 1

n ,1n

]) has no simple

Σ–presentations over HF(M).


The idea of the proof

Actually we use decompositions of kind

a =∑i<ω

ai ti ,

where t is an infinitesimal element.(Details are omitted)


Here we consider Hiblert spaces as structures of kind

H = 〈M,+, ·, (·, ·)〉

where M = H ] R,• H is the space itself• R is any set of cardinality 2ω (the set of reals without anyoperations and relations)• + is the predicate defining addition on H• · : R × H → H is the predicate defining the operation ofmultiplication by scalars• (·, ·) : H × H → R is the predicate defining the scalar product


Theorem

Any infinite dimensional Hilbert space has no simpleΣ–presentations over HF(M), for any ∃–Steinitz structure M.


Such a space is isomorphic to the space `2 consisting of sequencesthe reals (ai )i<ω satisfying the condition

∑i<ω a2

i <∞ withcoordinatewise operations and scalar product((a)i<ω, (b)i<ω

)=∑

i<ω aibiBy this we can assume H = `2. Basis:

(ei )(i) =

{1, if i = j

0, otherwise

Thus, (ai )i<ω =∑

i<ω aiei .Ai = [0, (i + 1)−1] · ei , Fi (x) = (x , ei ).For any sequence of values Fi (ai ), ai ∈ Ai take b =

∑i<ω(ai , ei )ei .

We have:

F (b) =

∑j<ω

(aj , ej)ej , ei

= (ai , ei ) = Fi (ai )


Some open questions:

Does Qp have a simple Σ–presentation over HF(R)?

Does R have a simple Σ–presentation over HF(Qp)?

Does there exist a nonstandard extension of R of cardinality2ω with a simple Σ–presentation over HF(R)?

The same questions without the requirement of simplicity ofpresentations.

Does any compatible countable theory have a model of thecardinality 2ω simply presentable over HF(R)? (Withoutsimplicity: yes).


Yu. L. Ershov. Σ–Definability of algebraic structures, pages 235–260. Handbook of Recursive Mathematics,

vol. 1 (Recursive Model Theory), ser. Studies in Logic and Foundations of Mathematics, Elsevier publ.,Amsterdam, Lausanne, New York, Oxford, Shannon, Singapore, Tokyo. 1998.

Yu.L. Ershov, V.G. Puzarenko, and A.I. Stukachev. HF–Computability. In S. Barry Cooper and Andrea

Sorbi, editors, Computability in Context. Computation and Logic in the Real World, pages 169–242,London, 2011. Imperial College Press.

A. S. Morozov. A Sufficient Condition for Nonpresentability of Structures in Hereditarily Finite

Superstructures. Algebra and Logic 2016, Vol. 55, No. 3, pages 242251.

A. S. Morozov. Nonpresenability of some structures of analysis in hereditarily finite superstructures. Algebra

and Logic, 2017 ? accepted.

A.S. Morozov One–dimensional Σ–presentations of structures over HF(R). In: infiniity, Computability, and

Metamathematics, College Publications, 2014, S. Geschke, B. Lowe, Philipp Schlicht, Vol. 23, TributesSeries, pages 285–298, London, Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch.


Thank you !


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