model theory of second order logic - lecture 1 · introduction second order characterizable...

IntroductionSecond order characterizable structures

Model theory of second order logicLecture 1

Jouko Väänänen1,2

1Department of Mathematics and Statistics, University of Helsinki

2ILLC, University of Amsterdam

August 2011

Jouko Väänänen Model theory of second order logic

Outline

1 IntroductionIntroductionNumber theoryAnalysisSet theory

2 Second order characterizable structuresSecond order characterizable structures

IntroductionNumber theoryAnalysisSet theory

Outline

Our framework

Usual set theoretic framework.In the third lecture a more general approach.

M = (M,R1, ...,Rn, f1, ..., fm, c1, ..., ck ).First order logic: for every element a of the domain of themodel ...Second order logic: for every subset of (or relation on) thedomain of the model ...Third order logic: for every family of subsets of (or relationbetween relations on) the domain of the model .......Sort logic: for every superset M ′ of the domain M andevery relation on M ′ ...

M = (M,R1, ...,Rn, f1, ..., fm, c1, ..., ck ).M is characterizable by a sentence (or theory) φ in somelogic if for allM′:

M′ ∼=M ⇐⇒ M′ |= φ.

M characterizable in first order logic iff M is finite.Infinite M: Cannot characterize in first order logic⇒ modeltheory.

M′ ∼=M ⇐⇒ M′ |= φ.

M = (M,R1, ...,Rn, f1, ..., fm, c1, ..., ck ).Can an infiniteM be characterized in second order logic?

For many: Yes.For all e.g. countable: Depends on set theory.Depends on model theoretic properties ofM.

Model theory of second order logic is very different frommodel theory of first order logic.

There is a more general notion of a structure: Henkinstructure (M,G), where G is a set of subsets of andrelations on M, for the range of second order quantifiers,such that certain comprehension axioms are satisfied.Every usual structureM gives rise to a Henkin structure(M,P(M) ∪ P(M2) ∪ ...), called a full structure.Some results about all full models hold also for Henkinmodels.Sometimes we cannot get a full model and have to settlewith a Henkin model.“Henkin model" is an auxiliary concept.Typically, full models “lurk" inside clouds of Henkin models.Truth in Henkin models can be axiomatized, truth in fullmodels not.

A rough correspondence

first order logic ∆KPU−Inf1 -part of set theory

first order logic with ∆1-part of set theorythe well-ordering quantifierfirst order logic with ∆1(Cd)-part of set theorythe Härtig quantifiersecond order logic ∆2-part of set theorysort logic set theoryBoolean valued sort Ω-logic(or second order) logic (Ongoing work with D. Ikegami)

Outline

Second order axioms of number theory, P2

Dedekind analyzed and Peano axiomatized number theory in1888 and 1889. Also Peirce 1881.

Constant symbol c for zero, function symbol S forsuccessor function.∀y¬Sy = c,∀y∀z(Sy = Sz → y = z),∀X ((Xc ∧ ∀y(Xy → XSy))→ ∀yXy) (Induction)

Second order axioms of number theory, P2

Theorem (Peirce 1881 (?), Dedekind 1888, Peano 1889 (?))

There is, up to isomorphism, only one model of P2, namely thestandard model (N,S,0).

Proof.

If (M, s,a) |= P2, interpret Y as a, sa, ssa, sssa, ....

In first order logic there is no theory T such that (N,S,0)is, up to isomorphism, the only model of T .Addition, multiplication, etc can be defined in second orderlogic from 0 and S, by recursion. Not so in first order logic.

Outline

An axiomatization of real numbers, R2

Karl Weierstrass brought the theory of the real numbers(from geometry) to algebra and set theory.David Hilbert axiomatized the ordered field of real numbersin 1902.

Real numbers, R2

Second order axioms of real numbers, R2:

Constant symbols for zero and one, function symbols +,×,relation symbol <.

• Field axioms for +, ·, 0 and 1• x < y → x + z < y + z• 0 < x ,0 < y → 0 < xy

• ∀X((∃yXy ∧ ∃x∀y(Xy → y ≤ x))

∃x∀y(∀z(Xz → z ≤ y)↔ x ≤ y

Characterization of the reals

Theorem

There is, up to isomorphism, only one model of R2, namely(R,+, ·,0,1,≤).

Proof.Given two models, find integers, then rationals in both. Maprationals to rationals. Extend to isomorphism usingcompleteness.

First order versus second order

In second order logic natural numbers can be defined onR. Not so in first order logic.Tarski 1948: The ordered field of reals is decidable in firstoder logic. Not so in second order logic.

Outline

Second order axioms of set theory, ZFC2, Zermelo1908

Usual ZFC axioms except replacement and separation arewritten in second order form. A finite set of axioms ZFC2

obtains.Models of ZFC2 are well-founded and the order-type oftheir ordinals is a strongly inaccessible cardinal.

Second order axioms of set theory, ZFC2

Theorem (Zermelo 1930)

For any strongly inaccessible cardinal > ω there is, up toisomorphism, exactly one model of ZFC2 with the set ofordinals of order-type κ, namely (Vκ,∈).

Proof.Transfinite induction on the rank. Suppose a well-founded(M,E) is a model of ZFC2 of inaccessible height κ. W.l.o.g.(M,E) is a transitive structure (M,∈), hence M ⊆ Vκ. Easily, Mis supertransitive (Suppose X ⊆ x ∈ M. Then X ⊆ M, as M is transitive. By

separation, X ∈ M). Hence M = Vκ.

In second order set theory mathematical questions outsideset theory are decided.Continuum Hypothesis CH ∀A ⊆ R(|A| ≤ |N| ∨ |A| = |R|) isdecided by ZFC2, but we do not (yet) know in which way ...... just as we do not (yet) know how (V ,∈) decides CH inset theory.

Second order characterizable structures

Outline

A model M is second order characterizable if there is asecond order sentence θM such that for all M ′:

M ′ |= θM ⇐⇒ M ′ ∼= M.

Characterizability by a theory is defined similarly.We already know many second order characterizablestructures. However, there are only countably many such.

M ′ |= θM ⇐⇒ M ′ ∼= M.

1 Natural numbers (N,+, ·, 0, 1, <)2 The field of real numbers (R,+, ·, 0, 1, <)3 The field of complex numbers (C,+, ·, 0, 1)4 The Euclidean space (Rn, || · ||)5 The Banach space (`p, || · ||p)6 The Hilbert space (Lp, || · ||p)7 The Euclidean geometry, consisting of points and lines8 The ordered set (ωn, <)9 The pure set (ℵn)

10 The Boolean algebra (P(ω),∩,∪, ∅,−)11 Any finite structure12 Any recursive countable structure13 The random graph14 The free group of ℵn generators15 The level of the cumulative hierarchy (Vω+n,∈)16 The level of the cumulative hierarchy (Vωn ,∈)17 The level of the cumulative hierarchy (Vκ,∈), where κ is the first inaccessible

(Mahlo, weakly compact, Ramsey) cardinal.18 Every consistent recursive first order theory has a second order characterizable

model in each second order characterizable infinite cardinality.

Non-second order characterizable structures

Not second order characterizable:(R,+,×,0,1, <,≺), where ≺ is a well-order, assuminglarge cardinals. (Martin-Steel 1989)(N, <,P), where P is the set of Gödel numbers of validsecond order sentences.

As to (N, <,P), we give a proof of a more general result:

Non-second order characterizable structures

Not second order characterizable:(R,+,×,0,1, <,≺), where ≺ is a well-order, assuminglarge cardinals. (Martin-Steel 1989)(N, <,P), where P is the set of Gödel numbers of validsecond order sentences.

As to (N, <,P), we give a proof of a more general result:

We give two proofs of:

TheoremThere is no second order characterizable structureM such thatthe set of Gödel numbers of valid second order sentences isTuring-reducible to truth inM.

Proof.The theory of any second order characterizable structure is ∆2(see below). The set of Gödel numbers of valid second ordersentences is Π2-complete (see below). A Π2-complete setcannot be Turing reducible to a ∆2-set, by the HierarchyTheorem of the Levy-hierarchy.

Figure: The hierarchy of second order characterizable structures

We prove two fact that we used in the above proof:

Theorem

If A is a second order characterizable structure, then the theoryof A is ∆2-definable.

TheoremThe set of second order φ such that |= φ, is Π2-complete.

Proof elements

L a finite vocabulary, A a second order characterizableL-structure.σ = the conjunction of a large finite part of ZFC.Sut(M) a Π1-formula which says that M is supertransitive.Voc(x) = the definition of “x is a vocabulary".SO(L, x) = the set-theoretical definition of the class ofsecond order L-formulas.Str(L, x) = the set-theoretical definition of L-structures.Sat(A, φ) = the inductive truth-definition of second orderlogic written in the language of set theory.

Two ways to say y is true in the z-structure x

P1(z, x , y) = Voc(z) ∧ Str(z, x) ∧ SO(z, y) ∧∃M(z, x , y ∈ M ∧ σ(M) ∧ Sut(M)

∧(Sat(z, x , y))(M)) (Σ2)

P2(z, x , y) = Voc(z) ∧ SO(z, y) ∧ Str(z, x) ∧∀M((z, x , y ∈ M ∧ σ(M) ∧ Sut(M))

→ (Sat(z, x , y))(M)) (Π2).

ZFC ` ∀z∀x∀y(P1(z, x , y)↔ P2(z, x , y))

A |= φ iff ∃x(P1(L, x , φ) ∧ P1(L, x , θA)) iff∀x(P1(L, x , θA)→ P2(L, x , φ)). This shows that the secondorder theory of a second order characterizable A is ∆2.If L is a vocabulary and φ a second order L-sentence, then|= φ ⇐⇒ ∀x(Str(L, x)→ P1(L, x , φ)). This shows that “φis valid" is Π2. Now we show it is Π2-complete.

Suppose ∃x∀yP(x , y ,n) is a Σ2-predicate. Let φn be asecond order sentence the models of which are, up toisomorphism, exactly the models (Vα,∈), where α = iαand (Vα,∈) |= ∃x∀yP(x , y ,n). If ∃x∀yP(x , y ,n) holds, wecan find a model for φn by means of the Levy Reflectionprinciple. On the other hand, suppose φn has a model.W.l.o.g. it is of the form (Vα,∈). Let a ∈ Vα such that(Vα,∈) |= ∀yP(a, y ,n). Since in this case Hα = Vα,(Hα,∈) |= ∀yP(a, y ,n), where Hα is the set of sets ofhereditary cardinality < α. By another application of theLevy Reflection Principle we get (V ,∈) |= ∀yP(a, y ,n), andwe have proved ∃x∀yP(x , y ,n).

model theory of second order logic - lecture 1 · introduction second order characterizable...

Documents