Saturated models and disjunctions insecond-order arithmetic

David Belanger

6 October 2013At Dartmouth College

EMAIL: dbelanger@math.cornell.eduWEB: http://www.math.cornell.edu/∼dbelange

Department of MathematicsCornell University

Existence of saturated models.

A countable saturated model of a theory T is one that realizes alltypes with finitely many parameters.

Theorem (Classical.)

Every complete consistent countable theory with only countablymany types has a countable saturated model.

Why?

Beginning with the most obvious:

1. To study the reverse mathematics of classical model theory.(Because it’s there.)

2. To find the dividing line between classical and effective modeltheory.

3. To find new degrees of reverse-mathematical strength.

New degrees of reverse-math strength.

For example:

1. AMT (Hirschfeldt, Shore, Slaman 08)

2. Π01G (Ibid.)

3. Π01GA (Hirschfeldt, Lange, Shore TA)

4. ACA0 ∨ ¬WKL0 (Belanger TA)

5. WKL0 ∨ IΣ02 (This talk)

4 and 5 come from theorems which have both a classically validproof using comprehension axioms, and an effective proof in theω-model REC.

QuestionIf RCA0 ` (WKL0 ∨ IΣ0

2)↔ P, what does P’s best proof looklike?

Reverse mathematics.

We consider three of the ‘Big Five’ subsystems of second-orderarithmetic:

ACA0 Arithmetic Comprehension Axiom0′ exists

WKL0 Weak Konig’s Lemmaa PA degree exists

RCA0 Recursive Comprehension Axiomcomputable sets exist

as well as

IΣ02 Induction principle for Σ0

2 formulas(φ(0) ∧ ∀n(φ(n)→ φ(n + 1)))→ ∀nφ(n), where φ is Σ0

2

BΣ02 Bounding principle for Σ0

2 formulas.We’ll talk about it later.

Basic model theory.

• Type-omitting

• Countable homogeneous models

• Countable saturated models

• Elementary embeddings...

• Material from an introductory course.

And everything is countable.

Some definitions are easy to formalize.

Work in a model (M,S) of RCA0, where M is the first-order part,and S is the second-order part.

• A theory is a set T ∈ S of first-order sentences over somelanguage L ∈ S.

• A model A ∈ S of T is an elementary diagram containing T .

• An n-type p ∈ S of T is a maximal set of n-ary formulasconsistent with T .

• An n-type p of T is principal if there is a φ ∈ p such that noother n-type of T contains φ.

• Two models A,B ∈ S are isomorphic if there is an f ∈ Swhich is an isomorphism between them.

Some theorems are easy to formalize.

Some known results:

Theorem (Completeness; RCA0)

Every complete consistent theory has a model.

Theorem (Compactness; WKL0)

Every finitely satisfiable theory has a model.

Theorem (Type Omitting; RCA0)

If T is a complete consistent theory and p is a nonprincipal type ofT , then T has a model which omits p.

Some things are less clear-cut.

The following are classically equivalent:

• A countable model is atomic if it realizes only principal types.

• A countable model A of T is prime if it embeds elementarilyinto every model.

This equivalence has the strength of ACA0 over RCA0

(Hirschfeldt, Shore, Slaman 09).

Another related pair:

• A countable model of T is saturated if it realizes every typewith parameters.

• A countable model A of T is universal if every countablemodel embeds elementarily into it.

Classically, every saturated model is universal. This implication hasthe strength of ACA0 over RCA0 (Harris 06).

Some things are even worse.

Fix a structure A.

If for every pair a, b of tuples such that tpA(a) = tpA(b) . . .

• . . .and every element u there is a v such thattpA(au) = tpA(bv), then A is 1-point homogeneous.

• . . .and every tuple u there is a tuple v such thattpA(au) = tpA(bv), then A is 1-homogeneous.

• . . .there is an automorphism of A taking a to b pointwise,then A is strongly 1-homogeneous.

These are all equivalent in classical mathematics.

No two are provably equivalent in RCA0 (Hirschfeldt, Lange,Shore TA):

• 1-point homogeneous ⇔ 1-homogeneous is equivalent to IΣ02.

• strongly 1-homogeneous ⇔ the other two is equivalent toACA0.

Existence of homogeneous models.

Theorem (Classical.)

Every complete consistent countable theory has a countablehomogeneous model.

Theorem (Essentially Lange 08)

TFAE over RCA0:

1. WKL0

2. Every complete consistent theory has a 1-point homogeneousmodel.

3. Every complete consistent theory has a 1-homogeneous model.

4. Every complete consistent theory has a strongly1-homogeneous model.

Existence of saturated models.

Theorem (Classical.)

Every complete consistent countable theory with only countablymany types has a saturated model.

DefinitionA complete theory T has countably many types if there is asequence 〈p0, . . .〉 such that

• each pi is a type of T ; and

• each type of T is equal to some pi .

Theorem (1st version.)

RCA0 `WKL0 ↔ Every complete theory with countably manytypes has a saturated model.

RCA0 `WKL0 ← (∀ complete theory with ℵ0-many types∃ saturated model).

Proof.Alter a construction of Millar (79) to produce a theory with twodecidable nonprincipal 1-types p, q such that any model realizingboth has PA degree. When carried out in a model ofRCA0 + ¬WKL0, this gives a theory with types p, q such that nomodel realizes both p and q.

(More on Millar’s construction later.)

RCA0 `WKL0 → (∀ complete theory with ℵ0-many types∃ saturated model).

DefinitionA model is ∅-saturated if it realizes every type (withoutparameters).

LemmaRCA0 ` A model is saturated iff it is ∅-saturated and1-homogeneous.

Proof of Theorem.Fix a model (M,S) of WKL0, and T ∈ S with an enumeration〈p0, . . .〉 of all its types.Build a ∅-saturated, 1-homogeneous model by a Henkin-styleconstruction.

Proof (continued).

Idea:

• In a Henkin construction, there are stages where we may addeither φ or ¬φ to the diagram.

• Represent the possible choices as a binary tree H.

• Prune H to an infinite subtree H∗ where every path encodes a∅-saturated, 1-homogeneous model.

• Then WKL0 gives us the model.

Proof (continued).

More formal:Let L = the language, L′ = L ∪ {c0, . . .}, each ci a new constant,and (φs)s∈M a list of all L′ sentences. Build a binary tree H by:

H∅ = ∅Hσ0 = Hσ ∪ ¬φ|σ|Hσ1 = Hσ ∪ φ|σ| ∪ (assign a Henkin witness c2k+1)

H = {σ ∈ 2<M : T ∪ Hσ is consistent}

Notice:

• H is infinite.

• Each path in H encodes a model of T .

• We only really mess with odd-indexed c2k+1.

Proof (continued).

H = {σ ∈ 2<M : T ∪ Hσ is consistent}

Use the even-indexed c2k to define sets Φhom,s and Φsat,s ofsentences such that

Hhom = {σ ∈ 2<M : T ∪ Hσ ∪ Φhom,|σ| is consistent},

Hsat = {σ ∈ 2<M : T ∪ Hσ ∪ Φsat,|σ| is consistent},

are infinite subtrees of H, and:

• Any path of Hhom encodes a 1-homogeneous model of T .

• Any path of Hsat encodes a ∅-saturated model of T .

• Hhom ∩Hsat is an infinite tree.

Then we’re done!

Existence of saturated models, again.The proof of ← used types which could not be amalgamated.

DefinitionA theory has pairwise type amalgamation if its types obey the law:

q0(x , y) � v

))

p(x))

77

� u

''

∃r(x , y , z)

q1(x , z)( �

55

Fact:If T has a saturated model, T has pairwise type amalgamation.

Theorem (2nd version.)

RCA0 + BΣ02 ` (WKL0 ∨ IΣ0

2)↔ Every complete theory withcountably many types and with pairwise type amalgamation has asaturated model.

RCA0 + BΣ02 ` (WKL0 ∨ IΣ0

2)→ Ctbly many types andp.w. type amalgamation implies there is a saturated model.

Proof.Show separately that WKL0 and RCA0 + IΣ0

2 each imply theconclusion.

• WKL0 does it by the Henkin tree construction we saw earlier.

• IΣ02 does it by a finite injury argument. (Namely, a Henkin

construction which tries to assign a witness to every type withparameters.)

The ← direction requires more explanation.

Two lemmas.

Define an n-wise type amalgamation property analogously with thepairwise:

q0(x , y (0)) � y,,� |--

p(x)' �

44

� y

++

# �22

� � // � � // ∃r(x , y (0), . . . , y (n−1))

qn−1(x , y (n−1)z)# � 22

LemmaRCA0 + BΣ0

2 ` If a complete theory T has a saturated model,then it has n-wise type amalgamation for all n.

LemmaRCA0 ` (WKL0 ∨ IΣ0

2)↔ (Ctbly many types and pairwise typeamalgamation implies (∀n)n-wise type amalgamation).

Millar’s Construction.

Fact:There is a pair U,V of c.e. sets such that U ⊆ C ⊆ (N− V )implies C has PA degree.

Let L = {Ps unary,Rs binary : s ∈ N}. Millar’s L-theory T :

• If A |= T and A |= Ps(a), say a is turned on at stage s.

• Every a ∈ A is turned on at a set of the form [0, t),t ∈ {0, . . . , ω}.

• If a 6= b are both turned on at stage s, then

U � s ⊆ {t < s : A |= Rt(a, b)} ⊆ (N− V � s).

T has a computable nonprincipal 1-type p(x) =‘x is turned on atevery stage’. If two elements a 6= b each realize p, the set

C = {t : A |= Rt(a, b)}has PA degree.

RCA0 ` (Ctbly many types and pairwise typeamalgamation implies (∀n)n-wise type

amalgamation)→ (WKL0 ∨ IΣ02).

In (M,S) |= RCA0 + ¬WKL0, Millar’s T is a complete theorywith a nonprincipal 1-type p which is never realized twice.Modify so that in (M,S) |= RCA0 + ¬WKL0 + ¬IΣ0

2 we get:

• A complete theory T .

• T has countably many types.

• T has pairwise type amalgamation.

• A tuple 〈p0(x0), . . . , pn−1(xn−1)〉 of 1-types.

• No n-type extends p0(x0) ∪ · · · ∪ pn−1(xn−1).

Summary of the tricks.

LemmaTFAE over RCA0:

1. IΣ02

2. If D1 ⊆ D2 ⊆ · · · is a sequence of sets, D1 is finite, and Dn

finite implies Dn+1 finite, then all Dn are finite.

3. If D1 ⊆ D2 ⊆ · · · is a sequence of sets, D1 is finite, and Dn

finite implies D2n finite, then all Dn are finite.

In (M,S) |= RCA0 + ¬WKL0 + ¬IΣ02:

• Fix a counterexample D1 ⊆ · · · to 3, say with DN infinite.

• Let L = (Ps unary, Rks k-ary : s ∈ M, k < N).

• Define T so that:

• Each Rks tries to separate initial segments of U from V .

• Rks (a0, . . . , ak−1) holds only if every ai is turned on at stage s,

and only if s ∈ Dk .

To summarize.

LemmaRCA0 ` (WKL0 ∨ IΣ0

2)↔ (If T has pairwise type amalgamation,T has n-wise type amalgamation for all n).

Theorem (2nd version.)

RCA0 + BΣ02 ` (WKL0 ∨ IΣ0

2)↔ Every complete theory withcountably many types and with pairwise type amalgamation has asaturated model.

QuestionIf RCA0 ` (WKL0 ∨ IΣ0

2)↔ P, what does P’s best proof looklike?

References

• S.G. Simpson. Subsystems of second-order arithmetic,Perspectives in Logic, 2009.

• K. Lange. The computational complexity of homogeneousmodels, doctoral dissertation, U. Chicago, 2008.

• D. Hirschfeldt, R. Shore, T. Slaman, ‘The atomic modeltheorem and type omitting,’ TAMS, 2009.

• D. Hirschfeldt, K. Lange, R. Shore, ‘Induction, bounding,weak combinatorial principles, and the homogeneous modeltheorem,’ to appear.

• K. Harris, ‘Reverse mathematics of saturated models,’unpublished, 2006. Availablehttp://kaharris.org/papers/reverse-sat.pdf.

• D. Belanger, ‘Reverse mathematics of first-order theories withfinitely many models,’ to appear.

• D. Belanger, ‘WKL0 and induction principles in modeltheory,’ to appear.