Many-Sorted First-Order Model Theory Lecture 11 16 th July, 2020 1 / 33
Homogeneity and ω -categoricity 2 / 33
Theories with “essentially only one” model ◮ Easy case: theories with no infinite models. Such a theory can have only one finite model (up to isomorphism). ◮ For theories that have infinite models, by the upward L¨ owenheim-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. 3 / 33
Theories with “essentially only one” model ◮ Easy case: theories with no infinite models. Such a theory can have only one finite model (up to isomorphism). ◮ For theories that have infinite models, by the upward L¨ owenheim-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. 4 / 33
Theories with “essentially only one” model ◮ Easy case: theories with no infinite models. Such a theory can have only one finite model (up to isomorphism). ◮ For theories that have infinite models, by the upward L¨ owenheim-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. 5 / 33
Theories with “essentially only one” model ◮ Easy case: theories with no infinite models. Such a theory can have only one finite model (up to isomorphism). ◮ For theories that have infinite models, by the upward L¨ owenheim-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. 6 / 33
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¨ owenheim-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. 7 / 33
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¨ owenheim-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. 8 / 33
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. 9 / 33
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. 10 / 33
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. 11 / 33
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. 12 / 33
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. 13 / 33
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. 14 / 33
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. 15 / 33
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