20 Years of Tameness

Monica VanDieren

March 8, 2022

20 Years of Tameness +

Honors Program

1

20 Years of Tameness +

2

Agenda

• Classification Theory for first order logic

• Non-first order contexts - including Tame AECs

• ​Why Study Tame AECs?

  • Ubiquitous

  • Informs Shelah's Categoricity Conjecture

  • New Model Theoretic Ideas (e.g. limit models)

• ​Classification Theory for Tame AECs

3

Morley's Theorem: Suppose \(T\) is a countable first order theory. If there exists an uncountable \(\lambda\) such that \(T\) is \(\lambda\)-categorical, then \(T\) is \(\mu\)-categorical in all uncountable \(\mu\).

4

\(\aleph_0\)

\(\aleph_1\)

\(\aleph_0\)

categorical in \(\lambda\)

Morley's Theorem: Suppose \(T\) is a countable first order theory. If there exists an uncountable \(\lambda\) such that \(T\) is \(\lambda\)-categorical, then \(T\) is \(\mu\)-categorical in all uncountable \(\mu\).

4

\(\aleph_0\)

\(\aleph_1\)

\(\aleph_0\)

categorical in \(\lambda\)

categorical in \(\mu\)

uncountably categorical

Theory of algebraically closed fields of fixed characteristic (Steinitz, 1910)

Classifying First Order (countable) Theories

All complete first order theories of countable signature

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uncountably categorical

Theory of multicolored directed graphs omitting directed cycles 

Theory of free groups on n>1
generators 

Theory of differentially closed fields of characteristic 0 

Theory of algebraically closed fields of fixed characteristic (Steinitz, 1910)

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Theory of random graph

5

uncountably categorical

Theory of multicolored directed graphs omitting directed cycles (Shelah, 1996)

Theory of free groups on n>1
generators (Sela, 2006)

Theory of differentially closed fields of characteristic 0 (Blum, 1968)

Theory of algebraically closed fields of fixed characteristic (Steinitz, 1910)

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Theory of random graph

superstable

stable

simple

NSOP

5

uncountably categorical

Classifying First Order (countable) Theories

All complete first order theories of countable signature

superstable

stable

simple

NSOP

6

uncountably categorical

Classifying First Order (countable) Theories

All complete first order theories of countable signature

superstable

stable

simple

NSOP

Increasing levels of structure

(e.g. independence relations,
rank functions, etc.)

6

Can we develop a similar classification for non-first order theories?

?

7

First Order  Theories

8

First Order Theories

\(L_{\omega_1,\omega}\) Theories

Incomplete Map of Non-elementary Classes

8

First Order Theories

\(L_{\omega_1,\omega}\) Theories

Incomplete Map of Non-elementary Classes

8

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Incomplete Map of Non-elementary Classes

8

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Abstract Elementary Classes

Incomplete Map of Non-elementary Classes

8

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Abstract Elementary Classes

Tame

Incomplete Map of Non-elementary Classes

8

Abstract Elementary Classes

Classification Theory for

Tame

9

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

9

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Stable

3

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Stable

9

Test Question for
Classification Theory

Morley's Theorem: Suppose \(T\) is a countable first order theory. If there exists an uncountable \(\lambda\) such that \(T\) is \(\lambda\) categorical, then \(T\) is \(\mu\)-categorical in all uncountable \(\mu\).

10

Test Question for
Classification Theory

Infinitary Logic Conjecture (Shelah, 1976): Suppose \(\psi\in L_{\omega_1,\omega}\) in a countable language. If there exists a \(\lambda\geq\beth_{\omega_1}\) such that \(\psi\) is \(\lambda\) categorical, then \(\psi\) is \(\mu\)-categorical in all \(\mu\geq \beth_{\omega_1}\).

Morley's Theorem: Suppose \(T\) is a countable first order theory. If there exists an uncountable \(\lambda\) such that \(T\) is \(\lambda\) categorical, then \(T\) is \(\mu\)-categorical in all uncountable \(\mu\).

10

Test Question for
Classification Theory

Infinitary Logic Conjecture (Shelah, 1976): Suppose \(\psi\in L_{\omega_1,\omega}\) in a countable language. If there exists a \(\lambda\geq\beth_{\omega_1}\) such that \(\psi\) is \(\lambda\) categorical, then \(\psi\) is \(\mu\)-categorical in all \(\mu\geq \beth_{\omega_1}\).

AEC Conjecture: Suppose \(\mathcal{K}\) is an AEC in a language of cardinality \(LS(\mathcal{K})\).  If there exists a \(\lambda \geq \beth_{(2^{LS(\mathcal{K})})^+}\)\(^*\) such that  \(\mathcal{K}\)  is \(\lambda\)-categorical, then   \(\mathcal{K}\) ​ is \(\mu\)-categorical in all \(\mu\geq \beth_{(2^{LS(\mathcal{K})})^+}\).

Morley's Theorem: Suppose \(T\) is a countable first order theory. If there exists an uncountable \(\lambda\) such that \(T\) is \(\lambda\) categorical, then \(T\) is \(\mu\)-categorical in all uncountable \(\mu\).

\(^*\) This bound is smaller when \(LS(\mathcal{K})=\aleph_0\).

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AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a ordering, \(\prec\), satisfying:

11

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

If \(M\in\mathcal{K}\) and \(M\cong N\), then \(N\in\mathcal{K}\).

11

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

12

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

If \(M,N\in\mathcal{K}\) and \(M\prec N\),

then \(M\) is a submodel of \(N\).

12

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^*\), \(N\prec M^*\), and \(M\subseteq N\),

then \(M\prec N\)

13

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^*\), \(N\prec M^*\), and \(M\subseteq N\),

then \(M\prec N\)

\(M\)

\(M^*\)

13

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^*\), \(N\prec M^*\), and \(M\subseteq N\),

then \(M\prec N\)

\(N\)

\(M^*\)

13

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^*\), \(N\prec M^*\), and \(M\subseteq N\),

then \(M\prec N\)

\(M\)

\(N\)

\(M^*\)

13

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^*\), \(N\prec M^*\), and \(M\subseteq N\),

then \(M\preceq N\).

\(M\)

\(N\)

\(M^*\)

13

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

14

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(A\)

\(M\)

A4. Löwenheim Skolem

There exists \(\lambda = LS(\mathcal{K})\) such that for every \(M\in\mathcal{K}\) and for every \(A\subseteq M\), there exists \(N\in\mathcal{K}\) so that \(N\prec M\) and \(\|N\|\leq |A|+\lambda\).

14

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(A\)

\(N\)

\(M\)

A4. Löwenheim Skolem

There exists \(\lambda=LS(\mathcal{K})\) such that for every \(M\in\mathcal{K}\) and for every \(A\subseteq M\), there exists \(N\in\mathcal{K}\) so that \(N\preceq M\) and \(\|N\|\leq |A|+\lambda\).

14

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5a. Union Conditions

15

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence, then \(\bigcup_{i<\alpha}M_i\in\mathcal{K}\) and for each \(i<\alpha\), \(M_i\prec \bigcup_{i<\alpha}M_i\).

A5a. Union Conditions

. . .

. . .

15

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence, then \(\bigcup_{i<\alpha}M_i\in\mathcal{K}\) and for each \(i<\alpha\), \(M_i\prec \bigcup_{i<\alpha}M_i\).

A5a. Union Conditions

. . .

. . .

\(\bigcup_{i<\alpha}M_{i}\)

15

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_i\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence, then \(\bigcup_{i<\alpha}M_i\in\mathcal{K}\) and for each \(i<\alpha\), \(M_i\prec \bigcup_{i<\alpha}M_i\).

A5a. Union Conditions

\(\bigcup_{i<\alpha}M_{i}\)

\(M_0\)

\(M_{i+1}\)

. . .

. . .

15

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5a.

A5b. Union Conditions

16

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence and \(N\) is such that for each \(i<\alpha\), \(M_i\prec N\), then \(\bigcup_{i<\alpha}M_i\prec N\).

. . .

. . .

A5a.

A5b. Union Conditions

16

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence and \(N\) is such that for each \(i<\alpha\), \(M_i\prec N\), then \(\bigcup_{i<\alpha}M_i\prec N\).

A5b. Union Conditions

. . .

. . .

\(N\)

A5a.

16

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence and \(N\) is such that for each \(i<\alpha\), \(M_i\prec N\), then \(\bigcup_{i<\alpha}M_i\preceq N\).

A5b. Union Conditions

. . .

. . .

\(\bigcup_{i<\alpha}M_{i}\)

\(N\)

A5a.

16

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M_1\)

17

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(M_1\)

17

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(f\)

\(M_1\)

17

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(f\)

\(f(M_1)\)

\(M_1\)

17

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(f\)

\(f(M_1)\)

\(M_1\)

\(f\restriction N=id_N\)

17

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

Joint Embedding Property

\(M_1\)

\(M_2\)

18

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

Joint Embedding Property

\(M_1\)

\(M_2\)

\(M'\)

18

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

Joint Embedding Property

\(f_1(M_1)\)

\(f_2(M_2)\)

\(M'\)

\(f_1\)

\(f_2\)

\(M_1\)

\(M_2\)

AP

18

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

\(M\)

No Maximal Models

AP

JEP  

19

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

\(M\)

No Maximal Models

\(M'\)

AP

JEP  

19

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

No Maximal Models

This implies the existence of a monster model.

\(M\)

20

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

\(M\)

20

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

No Maximal Models

\(M'\)

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

\(f\)

\(M\)

20

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(M\)

No Maximal Models

\(M'\)

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

\(f\)

\(f(M')\)

\(f\restriction M=id_M\)

20

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types over models.

21

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\)

\(a\)

\(b\)

21

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\)

\(a\)

\(M_a\)

\(M_b\)

\(b\)

21

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\) iff there exists \(f\) an automophism of \(\frak{C}\)

\(a\)

\(M_a\)

\(M_b\)

\(f\)

\(b\)

21

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\) iff there exists \(f\) an automophism of \(\frak{C}\)
so that \(f\restriction A=id_A\) and

\(a\)

\(b\)

\(M_a\)

\(M_b\)

\(f\)

\(f\restriction A=id_A\)

21

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP

A5. Union Conditions

JEP  

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\) iff there exists \(f\) an automophism of \(\frak{C}\)
so that  \(f\restriction A=id_A\) and \(f(a)=b\).

\(a\)

\(b=f(a)\)

\(M_a\)

\(M_b\)

\(f\)

\(f(M_a)\)

\(f\restriction A=id_A\)

21

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(A\subseteq N\) of cardinality \(<\chi\), so that \(q_a\restriction A\neq q_b\restriction A\). 

\(\frak{C}\)

\(a\)

\(b\)

\(N\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\subseteq N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M\)

\(b\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(b=f_M(a)\)

\(f_M(N)\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M\)

\(f_M\)

\(f_M\restriction M=id_M\)

\(N\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M'\)

\(b\)

\(N\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(b=f_{M'}(a)\)

\(f_{M'}(N)\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M'\)

\(f_{M'}\)

\(f_{M'}\restriction M'=id_{M'}\)

\(N\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M''\)

\(b\)

\(N\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(f_{M''}(N)\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M''\)

\(f_{M''}\)

\(b=f_{M''}(a)\)

\(f_{M''}\restriction M''=id_{M''}\)

\(N\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(b\)

22

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

Definition (Grossberg-V) A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\). 

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(b=f_N(a)\)

\(f_N\)

\(f_N\restriction N=id_N\)

22

Why study tameness?

23

Why study tameness?

23

A1: Tame AECs are "Everywhere"

Tame

AECs

Why study tameness?

23

A1: Tame AECs are "Everywhere"

Tame

Homogeneous

Finitary

Excellent

Universal

AECs

Why study tameness?

23

A1: Tame AECs are "Everywhere"

Tame

Homogeneous

Finitary

Excellent

Universal

?

AECs

Why study tameness?

23

A1: Tame AECs are "Everywhere"

Tame

Homogeneous

Finitary

Excellent

Universal

?

AECs

• \(Mod(\psi)\) where \(\psi\in L_{\kappa,\omega}\) with \(\kappa\) strongly compact (Makkai-Shelah)
   
• Homogeneous Classes
   
• Finitary Classes
   
• Quasi-minimal class axiomatizing Schanuel's Conjecture (Zilber)
   
• Excellent classes (Kolesnikov-Grossberg)
   
• Universal classes (Boney)
   
• All AECs are tame iff there is class-many almost strongly compact cardinals (Boney, Boney-Unger)

24

Why study tameness?

A1: Tame AECs are "Everywhere"

A2: Tameness Informs Work on Shelah's Categoricity Conj.

\(^*\) This bound is smaller when \(LS(\mathcal{K})=\aleph_0\).

Until 2006 most related results

involved set theoretic assumptions.

Shelah's Categoricity Conjecture: Suppose \(\mathcal{K}\) is an AEC in a language of cardinality \(LS(\mathcal{K})\).  If there exists a \(\lambda \geq \beth_{(2^{LS(\mathcal{K})})^+}\)\(^*\) such that  \(\mathcal{K}\)  is \(\lambda\)-categorical, then   \(\mathcal{K}\) ​ is \(\mu\)-categorical in all \(\mu\geq \beth_{(2^{LS(\mathcal{K})})^+}\).

25

Why study tameness?

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

categoricity in \(\lambda^+\)

categoricity in \(\lambda^+\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

26

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

categoricity in \(\mu\)

categoricity in \(\lambda^+\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\lambda^+\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\mu\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\lambda^+\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\mu\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\mu\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\lambda^+\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\mu\)

Upward and Downward Bounds Improved (Vasey, 2017)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\mu\)

Upward and Downward Bounds Improved (Vasey, 2017)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

26

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

Upward and Downward Bounds Improved (Vasey, 2017)

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

27

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{(2^{LS(\mathcal{K})})^+}<\mu<\lambda^+\).

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

27

Categoricity for Universal Classes (Vasey 2017)

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\).  

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

27

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

Categoricity for Universal Classes (Vasey 2017)

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\).  

27

What the downward/upward results leave open...

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

Categoricity for Universal Classes (Vasey 2017)

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\).  

27

What the downward/upward results leave open...

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\). If \(\mathcal{K}\) is categorical in some \(\lambda>\beth_{\beth_{\omega_1}}\) then \(\mathcal{K}\) is categorical in all \(\lambda>\beth_{\beth_{\omega_1}}\).

\(\lambda^+\)

 \(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

Categoricity for Universal Classes (Vasey 2017)

27

A3: The AEC setting uncovers new model theoretic concepts

Why study Tame AECs?

Limit Models

Frames & \(\mu\)-splitting

Towers

28

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

A3: The AEC setting uncovers new model theoretic concepts

Why study Tame AECs?

Limit Models

Frames & \(\mu\)-splitting

Towers

28

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}_\mu\mid i<\theta\rangle\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

\(\bigcup_{i<\theta}M_{i}\)

29

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(M'\)

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}_\mu\mid i<\theta\rangle\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

\(\bigcup_{i<\theta}M_{i}\)

29

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(M'\)

\(f\)

\(f(M')\)

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}_\mu\mid i<\theta\rangle\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

\(\bigcup_{i<\theta}M_{i}\)

29

Limit Models

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}_\mu\mid i<\theta\rangle\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

\(M'\)

\(f\)

\(f(M')\)

\(f\restriction M_i=id_{M_i}\)

29

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC and fix \(\mu>LS(\mathcal{K})\).

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M=M^\alpha_0\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

30

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC and fix \(\mu>LS(\mathcal{K})\).

\(M=M^\theta_0=M^\alpha_0\)

\(M^\theta_i\)

\(M^\theta_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

30

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC and fix \(\mu>LS(\mathcal{K})\).

\(M=M^\theta_0=M^\alpha_0\)

\(M^\theta_i\)

\(M^\theta_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

\(f\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

30

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC with AP, JEP,  and NMM, and fix \(\mu>LS(\mathcal{K})\).

Case \(cf(\alpha)=cf(\theta)\): Back and forth construction produces \(f\).

Case \(cf(\alpha)\neq cf(\theta)\): Answer seems related to "superstability".

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

31

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Case \(cf(\alpha)=\theta\): 

Fix increasing and continuous chain \(\langle \alpha_i\mid i<\theta\rangle\) so that \(\lim_{i<\theta}\alpha_i=\alpha\)

32

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Case \(cf(\alpha)=\theta\): 

32

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Case \(cf(\alpha)=\theta\): 

32

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Case \(cf(\alpha)=\theta\): 

32

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Case \(cf(\alpha)=\theta\): 

32

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Case \(cf(\alpha)=\theta\): 

32

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Round 0:

Player I picks a point in \(M_0\) and
Player II, defines \(f_0:N_{\alpha_0}\rightarrow M_0\) to be the identity mapping.

33

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_{i+1}}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

Round \(i\):

Player I picks a point in \(M_0\) and
Player II, defines \(f_0:N_{\alpha_0}\rightarrow M_0\) to be the identity mapping.

Suppose the game has proceeded to stage \(i+1\). 

So Player II has found \(f_i:N_{\alpha_i}\rightarrow M_{i}\)

33

\(M_0=N_{\alpha_0}\)

\(N_{\alpha_i}\)

\(\bigcup_{i<\theta} N_{\alpha_i}\)

Case \(cf(\alpha)=\theta\): Round \(i+1\)

\(M_{i+1}\)

\(M_{i}\)

\(\bigcup_{i<\theta} M_i\)

\(N_{\alpha_{i+1}}\)

\(M_0=N_{\alpha_0}\)

34

\(M_0=N_{\alpha_0}\)