Uniqueness triples from the diamond axiom

Ari Meir Brodsky ©

Ariel University

11 Iyyar, 5778Thursday, April 26, 2018

Set Theory, Model Theory and ApplicationsBen-Gurion University of the Negev, Eilat Campus

This is joint work with Adi Jarden.This research was carried out with the assistance of the Center forAbsorption in Science, Ministry of Aliyah and Integration, State ofIsrael.

Preliminaries

Fix a vocabulary τ .

We will consider a class of models K for τ ,along with a partial-order relation K on K.K will be a refinement of the submodel relation ⊆.

Preliminaries

Fix a vocabulary τ .We will consider a class of models K for τ ,along with a partial-order relation K on K.

K will be a refinement of the submodel relation ⊆.

Preliminaries

Fix a vocabulary τ .We will consider a class of models K for τ ,along with a partial-order relation K on K.K will be a refinement of the submodel relation ⊆.

Galois types

For a model A ∈ K and a point b /∈ A, we want to be able todescribe the type of b over A, that is, the relationship between band A.

To do so, we fix an ambient model B ∈ K that contains b.That is, for A,B ∈ K with A K B and b ∈ B \ A, we write

tp(b/A,B)

for the Galois-type of b over A, where b is viewed as an element ofB.When should two triples be considered to have the same type?

Galois types

For a model A ∈ K and a point b /∈ A, we want to be able todescribe the type of b over A, that is, the relationship between band A.To do so, we fix an ambient model B ∈ K that contains b.That is, for A,B ∈ K with A K B and b ∈ B \ A, we write

tp(b/A,B)

for the Galois-type of b over A, where b is viewed as an element ofB.

When should two triples be considered to have the same type?

Galois types

For a model A ∈ K and a point b /∈ A, we want to be able todescribe the type of b over A, that is, the relationship between band A.To do so, we fix an ambient model B ∈ K that contains b.That is, for A,B ∈ K with A K B and b ∈ B \ A, we write

tp(b/A,B)

for the Galois-type of b over A, where b is viewed as an element ofB.When should two triples be considered to have the same type?

Galois types

DefinitionWe say that (A, b,B) and (A, c,C) have the same type if thereexist model D and embeddings f : B → D and g : C → D that areconstant on A and such that f (b) = g(c).(This is an equivalence relation provided the AmalgamationProperty holds.)

If there is a monster model M in which all models lie, then:(A, b,B) and (A, c,C) have the same type iff there is anautomorphism of M that fixes A and sends b to c.

Galois types

DefinitionWe say that (A, b,B) and (A, c,C) have the same type if thereexist model D and embeddings f : B → D and g : C → D that areconstant on A and such that f (b) = g(c).(This is an equivalence relation provided the AmalgamationProperty holds.)If there is a monster model M in which all models lie, then:(A, b,B) and (A, c,C) have the same type iff there is anautomorphism of M that fixes A and sends b to c.

What classes of models do we consider?DefinitionAnd abstract elementary class is a class K of models for τ , withrelation K, satisfying:I K and K respect isomorphisms;I K is a partial order that refines ⊆;I If 〈Mα | α < δ〉 is a K-increasing sequence, then

M0 K⋃Mα | α < δ ∈ K;

I If 〈Mα | α ≤ δ〉 is a K-increasing, continuous sequence andMα K N for all α < δ, then Mδ K N;

I If A ⊆ B ⊆ C , A K C , and B K C , then A K B;I There is a Lowenheim-Skolem-Tarski number for K: the first

infinite cardinal λ such that for every N ∈ K and every subsetZ ⊆ N, there is M ∈ K such that Z ⊆ M K N and|M| ≤ λ+ |Z |.

Examples of Abstract Elementary Classes

I Let T be a first-order theory. Denote K := M | M |= T.Define M K N iff M is an elementary submodel of N.Then (K,K) is an AEC.

I Let T be a first-order theory with π2 axioms, that is, axiomsof the form ∀x∃yϕ(x , y), where ϕ is quantifier-free.Denote K := M | M |= T.Then (K,⊆) is an AEC.

I The class of locally finite groups with the relation ⊆ is anAEC.

Examples of Abstract Elementary Classes

I Let T be a first-order theory. Denote K := M | M |= T.Define M K N iff M is an elementary submodel of N.Then (K,K) is an AEC.

I Let T be a first-order theory with π2 axioms, that is, axiomsof the form ∀x∃yϕ(x , y), where ϕ is quantifier-free.Denote K := M | M |= T.Then (K,⊆) is an AEC.

I The class of locally finite groups with the relation ⊆ is anAEC.

Examples of Abstract Elementary Classes

I Let T be a first-order theory. Denote K := M | M |= T.Define M K N iff M is an elementary submodel of N.Then (K,K) is an AEC.

I Let T be a first-order theory with π2 axioms, that is, axiomsof the form ∀x∃yϕ(x , y), where ϕ is quantifier-free.Denote K := M | M |= T.Then (K,⊆) is an AEC.

I The class of locally finite groups with the relation ⊆ is anAEC.

Equivalent amalgamations

When are two amalgamations essentially the same?

When you can amalgamate them together in such a way that thediagram commutes. . . .

Equivalent amalgamations

When are two amalgamations essentially the same?When you can amalgamate them together in such a way that thediagram commutes. . . .

Domination triples

DefinitionSuppose A,B ∈ Kλ with A K B, and b ∈ B \ A. We say that bdominates B over A and that (A, b,B) is a domination triple if forevery C ∈ Kλ such that A K C , and any two amalgamations(f D

1 , idC ,D) and (f E1 , idC ,E ) of B and C over A that are not

equivalent over A, if f D1 (b), f E

1 (b) /∈ C , thentp(f D

1 (b)/C ,D) 6= tp(f E1 (b)/C ,E ).

ExampleK = class of fieldsK = subfieldThen (A, b,B) is a dominating triple if B = cl(A ∪ b).

Domination triples

DefinitionSuppose A,B ∈ Kλ with A K B, and b ∈ B \ A. We say that bdominates B over A and that (A, b,B) is a domination triple if forevery C ∈ Kλ such that A K C , and any two amalgamations(f D

1 , idC ,D) and (f E1 , idC ,E ) of B and C over A that are not

equivalent over A, if f D1 (b), f E

1 (b) /∈ C , thentp(f D

1 (b)/C ,D) 6= tp(f E1 (b)/C ,E ).

ExampleK = class of fieldsK = subfieldThen (A, b,B) is a dominating triple if B = cl(A ∪ b).

Adding a non-forking relation

We want to add a non-forking relation ^ to our structures:

^(A,B, c,C), where A,B,C ∈ K with A K B K C andc ∈ C \ B.The relation needs be invariant under isomorphisms, and also undertype equivalence, and satisfy certain monotonicity requirements.If ^(A,B, c,C), we say that tp(c/B,C) does not fork over A.If ^(A,A, c,C), we say that tp(c/A,C) is a basic type, denotedtp(c/A,C) ∈ Sbs(A).

Adding a non-forking relation

We want to add a non-forking relation ^ to our structures:^(A,B, c,C), where A,B,C ∈ K with A K B K C andc ∈ C \ B.

The relation needs be invariant under isomorphisms, and also undertype equivalence, and satisfy certain monotonicity requirements.If ^(A,B, c,C), we say that tp(c/B,C) does not fork over A.If ^(A,A, c,C), we say that tp(c/A,C) is a basic type, denotedtp(c/A,C) ∈ Sbs(A).

Adding a non-forking relation

We want to add a non-forking relation ^ to our structures:^(A,B, c,C), where A,B,C ∈ K with A K B K C andc ∈ C \ B.The relation needs be invariant under isomorphisms, and also undertype equivalence, and satisfy certain monotonicity requirements.

If ^(A,B, c,C), we say that tp(c/B,C) does not fork over A.If ^(A,A, c,C), we say that tp(c/A,C) is a basic type, denotedtp(c/A,C) ∈ Sbs(A).

Adding a non-forking relation

We want to add a non-forking relation ^ to our structures:^(A,B, c,C), where A,B,C ∈ K with A K B K C andc ∈ C \ B.The relation needs be invariant under isomorphisms, and also undertype equivalence, and satisfy certain monotonicity requirements.If ^(A,B, c,C), we say that tp(c/B,C) does not fork over A.

If ^(A,A, c,C), we say that tp(c/A,C) is a basic type, denotedtp(c/A,C) ∈ Sbs(A).

Adding a non-forking relation

We want to add a non-forking relation ^ to our structures:^(A,B, c,C), where A,B,C ∈ K with A K B K C andc ∈ C \ B.The relation needs be invariant under isomorphisms, and also undertype equivalence, and satisfy certain monotonicity requirements.If ^(A,B, c,C), we say that tp(c/B,C) does not fork over A.If ^(A,A, c,C), we say that tp(c/A,C) is a basic type, denotedtp(c/A,C) ∈ Sbs(A).

Uniqueness triples

DefinitionSuppose A,B ∈ Kλ with A K B, and b ∈ B \ A. We say that(A, b,B) is a uniqueness triple if tp(b/A,B) ∈ Sbs(A) and forevery C ∈ Kλ such that A K C , and any two amalgamations(f D

1 , idC ,D) and (f E1 , idC ,E ) of B and C over A that are not

equivalent over A, it cannot be that both tp(f D1 (b)/C ,D) and

tp(f E1 (b)/C ,E ) do not fork over A.

ExampleEquivalence classes. . . .

Uniqueness triples

DefinitionSuppose A,B ∈ Kλ with A K B, and b ∈ B \ A. We say that(A, b,B) is a uniqueness triple if tp(b/A,B) ∈ Sbs(A) and forevery C ∈ Kλ such that A K C , and any two amalgamations(f D

1 , idC ,D) and (f E1 , idC ,E ) of B and C over A that are not

equivalent over A, it cannot be that both tp(f D1 (b)/C ,D) and

tp(f E1 (b)/C ,E ) do not fork over A.

ExampleEquivalence classes. . . .

Why are uniqueness triples important?

Uniqueness triples allow us to move from a good λ-frame to agood λ+-frame, and thus to deduce existence of models ofcardinality λ+++.

Prior results

Theorem (Shelah)Suppose that:

1. 2λ < 2λ+< 2λ++ ;

2. s is a good λ-frame;3. I(λ++,K) < µunif(λ++, 2λ) ∼ 2λ++

Then every basic triple can be extended to a uniqueness triple.

New result

Theorem (Main Theorem, B. & Jarden, 2018)Suppose that:

1. λ is an infinite cardinal such that ♦(λ+) holds;2. s = (K,K, Sbs,^) is a good λ-frame;3. A ∈ Kλ;4. p ∈ Sbs(A).

Then there exist models C ,D ∈ Kλ such that A ≺K C ≺K D andb ∈ D \ C such that:

1. (C ,D, b) is a uniqueness triple;2. tp(b/A,D) = p; and3. ^(A,C , b,D).

A more useful form of ♦Jensen (1972) introduced the ♦ axiom to predict subsets of κ.But usually what we want to guess are subsets of some structureof size κ, not necessarily sets of ordinals.Encoding the desired sets as sets of ordinals is cumbersome, anddistracts us from properly applying the guessing power of ♦.

Definition (B. & Rinot, 2017)♦−(Hκ) asserts the existence of a sequence 〈Ωβ | β < κ〉 ofelements of Hκ such that for every parameter z ∈ Hκ+ and everysubset Ω ⊆ Hκ, there exists an elementary submodel M≺FO Hκ+

with z ∈M, such that κM :=M∩ κ is an ordinal < κ andM∩ Ω = ΩκM .Here, Hθ denotes the collection of all sets of hereditary cardinalityless than θ.Proposition (B. & Rinot, 2017)For any regular uncountable cardinal κ, ♦(κ) ⇐⇒ ♦−(Hκ).

A more useful form of ♦Jensen (1972) introduced the ♦ axiom to predict subsets of κ.But usually what we want to guess are subsets of some structureof size κ, not necessarily sets of ordinals.Encoding the desired sets as sets of ordinals is cumbersome, anddistracts us from properly applying the guessing power of ♦.Definition (B. & Rinot, 2017)♦−(Hκ) asserts the existence of a sequence 〈Ωβ | β < κ〉 ofelements of Hκ such that for every parameter z ∈ Hκ+ and everysubset Ω ⊆ Hκ, there exists an elementary submodel M≺FO Hκ+

with z ∈M, such that κM :=M∩ κ is an ordinal < κ andM∩ Ω = ΩκM .Here, Hθ denotes the collection of all sets of hereditary cardinalityless than θ.

Proposition (B. & Rinot, 2017)For any regular uncountable cardinal κ, ♦(κ) ⇐⇒ ♦−(Hκ).

A more useful form of ♦Jensen (1972) introduced the ♦ axiom to predict subsets of κ.But usually what we want to guess are subsets of some structureof size κ, not necessarily sets of ordinals.Encoding the desired sets as sets of ordinals is cumbersome, anddistracts us from properly applying the guessing power of ♦.Definition (B. & Rinot, 2017)♦−(Hκ) asserts the existence of a sequence 〈Ωβ | β < κ〉 ofelements of Hκ such that for every parameter z ∈ Hκ+ and everysubset Ω ⊆ Hκ, there exists an elementary submodel M≺FO Hκ+

with z ∈M, such that κM :=M∩ κ is an ordinal < κ andM∩ Ω = ΩκM .Here, Hθ denotes the collection of all sets of hereditary cardinalityless than θ.Proposition (B. & Rinot, 2017)For any regular uncountable cardinal κ, ♦(κ) ⇐⇒ ♦−(Hκ).

Proof of Main Theorem

Sketch of the proof — on the board. . .

References

Ari Meir Brodsky and Adi Jarden.Uniqueness triples from the diamond axiom.Preprint, arXiv:1804.10952, April 2018.https://arxiv.org/abs/1804.10952

Ari Meir Brodsky and Assaf Rinot.A Microscopic approach to Souslin-tree constructions. Part I.Annals of Pure and Applied Logic, 168(11): 1949–2007, 2017.

Saharon Shelah.Classification Theory for Abstract Elementary Classes 2.Studies in Logic: Mathematical logic and foundations, CollegePublications, 2009.Saharon Shelah.Non-structure in λ++ using instances of WGCH.Chapter VII, in series Studies in Logic, volume 20, CollegePublications. Sh:838. arxiv:0808.3020.