Saturated models and disjunctions insecond-order arithmetic
David Belanger
6 October 2013At Dartmouth College
EMAIL: [email protected]: 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.