Many-Sorted First-Order Model Theory Lecture 11 16 th July, 2020 1 - - PowerPoint PPT Presentation

Many-Sorted First-Order Model Theory

Lecture 11 16th July, 2020

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Homogeneity and ω-categoricity

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Theories with “essentially only one” model

◮ Easy case: theories with no infinite models. Such a theory can have

• nly one finite model (up to isomorphism).

◮ For theories that have infinite models, by the upward L¨

• wenheim-Skolem Theorem this cannot happen.

◮ We must settle on a weaker property.

Definition 1

Let λ be a cardinal. A theory T is called λ-categorical if for any A, B ∈ Mod(T) with card(A) = λ = card(B), we have A ∼ = B. ◮ A structure C is called λ-categorical if Th(C) is λ-categorical. ◮ Cantor’s theorem from Lecture 10 can be reformulated as follows.

Example 2

The theory of dense linear orders without endpoints is ω-categorical.

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Theories with “essentially only one” model

◮ Easy case: theories with no infinite models. Such a theory can have

• nly one finite model (up to isomorphism).

◮ For theories that have infinite models, by the upward L¨

• wenheim-Skolem Theorem this cannot happen.

◮ We must settle on a weaker property.

Definition 1

Let λ be a cardinal. A theory T is called λ-categorical if for any A, B ∈ Mod(T) with card(A) = λ = card(B), we have A ∼ = B. ◮ A structure C is called λ-categorical if Th(C) is λ-categorical. ◮ Cantor’s theorem from Lecture 10 can be reformulated as follows.

Example 2

The theory of dense linear orders without endpoints is ω-categorical.

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Theories with “essentially only one” model

◮ Easy case: theories with no infinite models. Such a theory can have

• nly one finite model (up to isomorphism).

◮ For theories that have infinite models, by the upward L¨

• wenheim-Skolem Theorem this cannot happen.

◮ We must settle on a weaker property.

Definition 1

Let λ be a cardinal. A theory T is called λ-categorical if for any A, B ∈ Mod(T) with card(A) = λ = card(B), we have A ∼ = B. ◮ A structure C is called λ-categorical if Th(C) is λ-categorical. ◮ Cantor’s theorem from Lecture 10 can be reformulated as follows.

Example 2

The theory of dense linear orders without endpoints is ω-categorical.

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Theories with “essentially only one” model

◮ Easy case: theories with no infinite models. Such a theory can have

• nly one finite model (up to isomorphism).

◮ For theories that have infinite models, by the upward L¨

• wenheim-Skolem Theorem this cannot happen.

◮ We must settle on a weaker property.

Definition 1

Let λ be a cardinal. A theory T is called λ-categorical if for any A, B ∈ Mod(T) with card(A) = λ = card(B), we have A ∼ = B. ◮ A structure C is called λ-categorical if Th(C) is λ-categorical. ◮ Cantor’s theorem from Lecture 10 can be reformulated as follows.

Example 2

The theory of dense linear orders without endpoints is ω-categorical.

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ω-categorical theories

Lemma 3

Let T be a consistent theory. If T has no finite models and is λ-categorical for some infinite cardinal λ, then T is complete.

Proof.

◮ Let A be the unique model of T with card(A) = λ. Clearly, T ⊆ Th(A). ◮ We will show that T ⊇ Th(A). Let ϕ ∈ Th(A). Suppose ϕ / ∈ T. ◮ Then there is a model B of T such that B | = ¬ϕ. By assumption, B is infinite. ◮ By L¨

• wenheim-Skolem (up- or downward), there is a model C with card(C) = λ such that

C | = T ∪ {¬ϕ}. ◮ By λ-categoricity, C ∼ = A. Thus, A | = ϕ, ¬ϕ. Contradiction.

Theorem 4 (local finiteness)

Let C be an ω-categorical structure. Then, every finitely generated substructure of C is finite. The proof relies on the Omitting Types Theorem, which we did not cover.

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ω-categorical theories

Lemma 3

Let T be a consistent theory. If T has no finite models and is λ-categorical for some infinite cardinal λ, then T is complete.

Proof.

◮ Let A be the unique model of T with card(A) = λ. Clearly, T ⊆ Th(A). ◮ We will show that T ⊇ Th(A). Let ϕ ∈ Th(A). Suppose ϕ / ∈ T. ◮ Then there is a model B of T such that B | = ¬ϕ. By assumption, B is infinite. ◮ By L¨

• wenheim-Skolem (up- or downward), there is a model C with card(C) = λ such that

C | = T ∪ {¬ϕ}. ◮ By λ-categoricity, C ∼ = A. Thus, A | = ϕ, ¬ϕ. Contradiction.

Theorem 4 (local finiteness)

Let C be an ω-categorical structure. Then, every finitely generated substructure of C is finite. The proof relies on the Omitting Types Theorem, which we did not cover.

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Where to find ω-categorical theories?

Definition 5

A structure A is uniformly locally finite if there is a function f : ω → ω such that for every substructure B of A the following holds: ◮ if B is generated by a set X with card(X) ≤ n, then card(B) ≤ f (n).

Two good points of ω-categoricity

Let T be an ω-categorical theory.

• 1. T enjoys quantifier elimination.
• 2. The (unique) countable model A of T is uniformly locally finite.

◮ Point (2) above suggests ω-categorical theories should be found among theories of structures that enjoy some kind of uniformity. ◮ This is indeed the case, and the uniformity we look for is called homogeneity.

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Where to find ω-categorical theories?

Definition 5

A structure A is uniformly locally finite if there is a function f : ω → ω such that for every substructure B of A the following holds: ◮ if B is generated by a set X with card(X) ≤ n, then card(B) ≤ f (n).

Two good points of ω-categoricity

Let T be an ω-categorical theory.

• 1. T enjoys quantifier elimination.
• 2. The (unique) countable model A of T is uniformly locally finite.

◮ Point (2) above suggests ω-categorical theories should be found among theories of structures that enjoy some kind of uniformity. ◮ This is indeed the case, and the uniformity we look for is called homogeneity.

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Where to find ω-categorical theories?

Definition 5

A structure A is uniformly locally finite if there is a function f : ω → ω such that for every substructure B of A the following holds: ◮ if B is generated by a set X with card(X) ≤ n, then card(B) ≤ f (n).

Two good points of ω-categoricity

Let T be an ω-categorical theory.

• 1. T enjoys quantifier elimination.
• 2. The (unique) countable model A of T is uniformly locally finite.

◮ Point (2) above suggests ω-categorical theories should be found among theories of structures that enjoy some kind of uniformity. ◮ This is indeed the case, and the uniformity we look for is called homogeneity.

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Homogeneous structures

Definition 6

A structure A is homogeneous, if every isomorphism between finitely generated substructures of A extends to an automorphism of A.

Example 7

The structure (Q, <) is homogeneous.

Exercise 1

Prove that (Q, <) is homogeneous.

Example 8

A random graph is a countable graph (V , E) with the following property: (P) For every pair X, Y of finite disjoint subsets of V , there exists a vertex v with E(x, v) for every x ∈ X and ¬E(y, v) for every y ∈ Y . Let G be a countable graph satisfying (P). Then G is homogeneous.

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Homogeneous structures

Definition 6

A structure A is homogeneous, if every isomorphism between finitely generated substructures of A extends to an automorphism of A.

Example 7

The structure (Q, <) is homogeneous.

Exercise 1

Prove that (Q, <) is homogeneous.

Example 8

A random graph is a countable graph (V , E) with the following property: (P) For every pair X, Y of finite disjoint subsets of V , there exists a vertex v with E(x, v) for every x ∈ X and ¬E(y, v) for every y ∈ Y . Let G be a countable graph satisfying (P). Then G is homogeneous.

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Homogeneous structures

Definition 6

A structure A is homogeneous, if every isomorphism between finitely generated substructures of A extends to an automorphism of A.

Example 7

The structure (Q, <) is homogeneous.

Exercise 1

Prove that (Q, <) is homogeneous.

Example 8

A random graph is a countable graph (V , E) with the following property: (P) For every pair X, Y of finite disjoint subsets of V , there exists a vertex v with E(x, v) for every x ∈ X and ¬E(y, v) for every y ∈ Y . Let G be a countable graph satisfying (P). Then G is homogeneous.

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HP, JEP and AP

Definition 9

A class K of structures some of signature Σ has hereditary property (HP), if whenever B ∈ K and A ≤ B, then A ∈ K.

Definition 10

A class K of structures some of signature Σ has joint embedding property (JEP), if for every A, B ∈ K there exists a C ∈ K and embeddings f : A ֒ → C and g : B ֒ → C.

Definition 11

A class K of structures some of signature Σ has amalgamation property (AP), if for every A, B, C ∈ K, and embeddings f : C ֒ → A and g : C ֒ → B, there exists a D ∈ K and embeddings h: A ֒ → D and k : B ֒ → D, such that for every c ∈ C we have h(f (c)) = k(g(c)).

Exercise 2

Find a class K enjoying AP but not JEP.

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HP, JEP and AP

Definition 9

A class K of structures some of signature Σ has hereditary property (HP), if whenever B ∈ K and A ≤ B, then A ∈ K.

Definition 10

A class K of structures some of signature Σ has joint embedding property (JEP), if for every A, B ∈ K there exists a C ∈ K and embeddings f : A ֒ → C and g : B ֒ → C.

Definition 11

A class K of structures some of signature Σ has amalgamation property (AP), if for every A, B, C ∈ K, and embeddings f : C ֒ → A and g : C ֒ → B, there exists a D ∈ K and embeddings h: A ֒ → D and k : B ֒ → D, such that for every c ∈ C we have h(f (c)) = k(g(c)).

Exercise 2

Find a class K enjoying AP but not JEP.

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HP, JEP and AP

Definition 9

A class K of structures some of signature Σ has hereditary property (HP), if whenever B ∈ K and A ≤ B, then A ∈ K.

Definition 10

A class K of structures some of signature Σ has joint embedding property (JEP), if for every A, B ∈ K there exists a C ∈ K and embeddings f : A ֒ → C and g : B ֒ → C.

Definition 11

A class K of structures some of signature Σ has amalgamation property (AP), if for every A, B, C ∈ K, and embeddings f : C ֒ → A and g : C ֒ → B, there exists a D ∈ K and embeddings h: A ֒ → D and k : B ֒ → D, such that for every c ∈ C we have h(f (c)) = k(g(c)).

Exercise 2

Find a class K enjoying AP but not JEP.

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HP, JEP and AP

Definition 9

A class K of structures some of signature Σ has hereditary property (HP), if whenever B ∈ K and A ≤ B, then A ∈ K.

Definition 10

A class K of structures some of signature Σ has joint embedding property (JEP), if for every A, B ∈ K there exists a C ∈ K and embeddings f : A ֒ → C and g : B ֒ → C.

Definition 11

A class K of structures some of signature Σ has amalgamation property (AP), if for every A, B, C ∈ K, and embeddings f : C ֒ → A and g : C ֒ → B, there exists a D ∈ K and embeddings h: A ֒ → D and k : B ֒ → D, such that for every c ∈ C we have h(f (c)) = k(g(c)).

Exercise 2

Find a class K enjoying AP but not JEP.

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Age and homogeneity

Definition 12

Let K be any structure. The age of K, written age(K), is the set of all finitely generated substructures of K.

Example 13

Let K be a structure, and let K = age(K). Then, K has HP and JEP.

Exercise 3

Find a finite structure K such that age(K) does not have AP.

Example 14

Consider Q and Z as ordered sets. Then: ◮ age(Q) = age(Z) = {finite linearly ordered sets}. ◮ Z is not homogeneous, but Q is (exercise). In this sense, Q is a better “limit” of its age, than Z.

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Age and homogeneity

Definition 12

Let K be any structure. The age of K, written age(K), is the set of all finitely generated substructures of K.

Example 13

Let K be a structure, and let K = age(K). Then, K has HP and JEP.

Exercise 3

Find a finite structure K such that age(K) does not have AP.

Example 14

Consider Q and Z as ordered sets. Then: ◮ age(Q) = age(Z) = {finite linearly ordered sets}. ◮ Z is not homogeneous, but Q is (exercise). In this sense, Q is a better “limit” of its age, than Z.

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Age and homogeneity

Definition 12

Let K be any structure. The age of K, written age(K), is the set of all finitely generated substructures of K.

Example 13

Let K be a structure, and let K = age(K). Then, K has HP and JEP.

Exercise 3

Find a finite structure K such that age(K) does not have AP.

Example 14

Consider Q and Z as ordered sets. Then: ◮ age(Q) = age(Z) = {finite linearly ordered sets}. ◮ Z is not homogeneous, but Q is (exercise). In this sense, Q is a better “limit” of its age, than Z.

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Age and homogeneity

Definition 12

Let K be any structure. The age of K, written age(K), is the set of all finitely generated substructures of K.

Example 13

Let K be a structure, and let K = age(K). Then, K has HP and JEP.

Exercise 3

Find a finite structure K such that age(K) does not have AP.

Example 14

Consider Q and Z as ordered sets. Then: ◮ age(Q) = age(Z) = {finite linearly ordered sets}. ◮ Z is not homogeneous, but Q is (exercise). In this sense, Q is a better “limit” of its age, than Z.

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Anything that can be an age, is

Theorem 15 (Converse to Example 13)

Let Σ be a signature and let K be a nonempty finite or countable set of finitely generated Σ-structures which has HP and JEP. Then, K is the age

• f some finite or countable structure.

Proof.

◮ Enumerate K as (Ai : i < ω), possibly with repetitions. ◮ Put B0 = A0. Using JEP, construct a chain of embeddings A0 = B0 B1 B2 B3 A1 A2 A3 ◮ Define B =

i∈ω Bi. By construction, every Ai ∈ age(B), so K ⊆ age(B).

◮ Let C ≤ B be finitely generated, say by X. Then X ⊆ Bi for some i, and so C ≤ Bi. ◮ By construction, each Bi is isomorphic to some Aj, so C ≤ Aj. ◮ By HP we have C ∈ K, showing that age(B) ⊆ K.

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Anything that can be an age, is

Theorem 15 (Converse to Example 13)

Let Σ be a signature and let K be a nonempty finite or countable set of finitely generated Σ-structures which has HP and JEP. Then, K is the age

• f some finite or countable structure.

Proof.

◮ Enumerate K as (Ai : i < ω), possibly with repetitions. ◮ Put B0 = A0. Using JEP, construct a chain of embeddings A0 = B0 B1 B2 B3 A1 A2 A3 ◮ Define B =

i∈ω Bi. By construction, every Ai ∈ age(B), so K ⊆ age(B).

◮ Let C ≤ B be finitely generated, say by X. Then X ⊆ Bi for some i, and so C ≤ Bi. ◮ By construction, each Bi is isomorphic to some Aj, so C ≤ Aj. ◮ By HP we have C ∈ K, showing that age(B) ⊆ K.

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Fra¨ ıss´ e’s limits

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Fra¨ ıss´ e Theorem

Theorem 16 (Fra¨ ıss´ e)

Let Σ be an at most countable signature and let K be a nonempty at most countable set of finitely generated Σ-structures which has HP, JEP and

• AP. Then, there is a Σ-structure D (called Fra¨

ıss´ e limit of K), unique up to isomorphism, such that

• 1. D is at most countable,
• 2. K = age(D),
• 3. D is homogeneous.

Definition 17

A structure D is weakly homogeneous if it has the following property: ◮ if A, B are finitely generated substructures of D, and A ≤ B, and f : A ֒ → D is an embedding, then there is an embedding g : B ֒ → D that extends f . Clearly, if D is homogeneous, it is weakly homogeneous.

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Fra¨ ıss´ e Theorem

Theorem 16 (Fra¨ ıss´ e)

Let Σ be an at most countable signature and let K be a nonempty at most countable set of finitely generated Σ-structures which has HP, JEP and

• AP. Then, there is a Σ-structure D (called Fra¨

ıss´ e limit of K), unique up to isomorphism, such that

• 1. D is at most countable,
• 2. K = age(D),
• 3. D is homogeneous.

Definition 17

A structure D is weakly homogeneous if it has the following property: ◮ if A, B are finitely generated substructures of D, and A ≤ B, and f : A ֒ → D is an embedding, then there is an embedding g : B ֒ → D that extends f . Clearly, if D is homogeneous, it is weakly homogeneous.

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Uniqueness: weak homogeneity and embeddings

Lemma 18

Let C and D be at most countable. Suppose age(C) ⊆ age(D) and D is weakly homogeneous. Then, any embedding from a finitely generated substructure of C into D can be extended to an embedding of C into D; in particular, C is embeddable into D.

Proof by picture.

◮ Let f0 : A0 ֒ → D be an embedding of a finitely generated A0 ≤ C into D. ◮ We extend it to fω : C ֒ → D as follows. Write C as

i<ω Ai of a chain of finitely

generated substructures of C. Inductive step: Ai Ai+1 fi(Ai) copy of Ai+1 C D ◮ Note that a copy of Ai+1 is a substructure of D because age(C) ⊆ age(D).

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Uniqueness: weak homogeneity and embeddings

Lemma 18

Let C and D be at most countable. Suppose age(C) ⊆ age(D) and D is weakly homogeneous. Then, any embedding from a finitely generated substructure of C into D can be extended to an embedding of C into D; in particular, C is embeddable into D.

Proof by picture.

◮ Let f0 : A0 ֒ → D be an embedding of a finitely generated A0 ≤ C into D. ◮ We extend it to fω : C ֒ → D as follows. Write C as

i<ω Ai of a chain of finitely

generated substructures of C. Inductive step: Ai Ai+1 fi(Ai) copy of Ai+1 C D ◮ Note that a copy of Ai+1 is a substructure of D because age(C) ⊆ age(D).

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Uniqueness: weak homogeneity and homogeneity

Lemma 19

• 1. Let C, D be Σ-structures with age(C) = age(D). Assume C, D are

both at most countable and weakly homogeneous. If A is a finitely generated substructure of C and f : A ֒ → D is an embedding, then f extends to an isomorphism from C to D. In particular, C ∼ = D.

• 2. A finite or countable structure is weakly homogeneous if and only if it

is homogeneous.

Proof.

◮ Write C =

i∈ω Ci and D = i∈ω Di, each a countable union of a chain of finitely

generated substructures. We will define a chain of isomorphisms fi between finitely generated substructures of C and D, so that, for each n ◮ domain of f2n includes Cn, ◮ range of f2n+1 includes Dn. ◮ Then f =

i<ω fi is the isomorphism we want.

◮ The step from f0 to f1 will be illustrated next.

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Uniqueness: weak homogeneity and homogeneity

Lemma 19

• 1. Let C, D be Σ-structures with age(C) = age(D). Assume C, D are

both at most countable and weakly homogeneous. If A is a finitely generated substructure of C and f : A ֒ → D is an embedding, then f extends to an isomorphism from C to D. In particular, C ∼ = D.

• 2. A finite or countable structure is weakly homogeneous if and only if it

is homogeneous.

Proof.

◮ Write C =

i∈ω Ci and D = i∈ω Di, each a countable union of a chain of finitely

generated substructures. We will define a chain of isomorphisms fi between finitely generated substructures of C and D, so that, for each n ◮ domain of f2n includes Cn, ◮ range of f2n+1 includes Dn. ◮ Then f =

i<ω fi is the isomorphism we want.

◮ The step from f0 to f1 will be illustrated next.

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Uniqueness: proof completed

Proof.

◮ Inductive step (from n = 0 to n = 1) by picture: C0 f0(C0)

g(f0(C0))

D0 G C D ◮ Start by picking f0: an embedding of C0 into D. ◮ Let G be the substructure of D generated by f0(C0) ∪ D0. ◮ Let g(G) be the image of G inside C (solid lines). ◮ Take the embedding of g(f0(C0)) into C0. By weak homogeneity, it extends to an embedding, say h, of g(G) into C (dashed lines). ◮ Put f1 = g−1 ◦ h−1.

This ends the proof of uniqueness. Existence will be proved next week.

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Uniqueness: proof completed

Proof.

◮ Inductive step (from n = 0 to n = 1) by picture: C0 f0(C0)

g(f0(C0))

D0 G C D ◮ Start by picking f0: an embedding of C0 into D. ◮ Let G be the substructure of D generated by f0(C0) ∪ D0. ◮ Let g(G) be the image of G inside C (solid lines). ◮ Take the embedding of g(f0(C0)) into C0. By weak homogeneity, it extends to an embedding, say h, of g(G) into C (dashed lines). ◮ Put f1 = g−1 ◦ h−1.

This ends the proof of uniqueness. Existence will be proved next week.

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