Many-Sorted First-Order Model Theory Lecture 8 25 th June, 2020 1 / 31
Applications of ultraproducts 2 / 31
Fancy models via ultraproducts Example 1 (Growing chains) Let C n be the n -element chain. Let U be a non-principal filter on ω . Then, � n <ω C n / U is an infinite chain consisting of a copy of ω at the bottom, a copy of dual ω at the top, and uncountably many copies of Z in between. Example 2 (Non-standard natural numbers) Let U be a non-principal ultrafilter on ω . Consider N ω / U . It is elementarily equivalent to N . Take the element (1 , 2 , 3 , 4 , . . . ) / U . It is strictly greater than any standard natural number. Example 3 (Infinitesimals) Let U be a non-principal ultrafilter on ω . Consider R ω / U . It is elementarily equivalent to R . Take the element (1 , 1 2 , 1 3 , 1 4 , . . . ) / U . It is strictly greater than 0, yet strictly smaller than any standard real number. 3 / 31
Fancy models via ultraproducts Example 1 (Growing chains) Let C n be the n -element chain. Let U be a non-principal filter on ω . Then, � n <ω C n / U is an infinite chain consisting of a copy of ω at the bottom, a copy of dual ω at the top, and uncountably many copies of Z in between. Example 2 (Non-standard natural numbers) Let U be a non-principal ultrafilter on ω . Consider N ω / U . It is elementarily equivalent to N . Take the element (1 , 2 , 3 , 4 , . . . ) / U . It is strictly greater than any standard natural number. Example 3 (Infinitesimals) Let U be a non-principal ultrafilter on ω . Consider R ω / U . It is elementarily equivalent to R . Take the element (1 , 1 2 , 1 3 , 1 4 , . . . ) / U . It is strictly greater than 0, yet strictly smaller than any standard real number. 4 / 31
Fancy models via ultraproducts Example 1 (Growing chains) Let C n be the n -element chain. Let U be a non-principal filter on ω . Then, � n <ω C n / U is an infinite chain consisting of a copy of ω at the bottom, a copy of dual ω at the top, and uncountably many copies of Z in between. Example 2 (Non-standard natural numbers) Let U be a non-principal ultrafilter on ω . Consider N ω / U . It is elementarily equivalent to N . Take the element (1 , 2 , 3 , 4 , . . . ) / U . It is strictly greater than any standard natural number. Example 3 (Infinitesimals) Let U be a non-principal ultrafilter on ω . Consider R ω / U . It is elementarily equivalent to R . Take the element (1 , 1 2 , 1 3 , 1 4 , . . . ) / U . It is strictly greater than 0, yet strictly smaller than any standard real number. 5 / 31
Elementary classes and ultraproducts Definition 4 A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod ( T ). Lemma 5 If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption of GCH. Later Shelah proved it with no special assumptions. ◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject. Theorem 6 (Keisler, Shelah) Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that A I / U ∼ = B I / U. 6 / 31
Elementary classes and ultraproducts Definition 4 A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod ( T ). Lemma 5 If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption of GCH. Later Shelah proved it with no special assumptions. ◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject. Theorem 6 (Keisler, Shelah) Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that A I / U ∼ = B I / U. 7 / 31
Elementary classes and ultraproducts Definition 4 A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod ( T ). Lemma 5 If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption of GCH. Later Shelah proved it with no special assumptions. ◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject. Theorem 6 (Keisler, Shelah) Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that A I / U ∼ = B I / U. 8 / 31
Elementary classes and ultraproducts Definition 4 A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod ( T ). Lemma 5 If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption of GCH. Later Shelah proved it with no special assumptions. ◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject. Theorem 6 (Keisler, Shelah) Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that A I / U ∼ = B I / U. 9 / 31
Elementary classes characterised Theorem 7 A ≡ B if and only if there is an elementary embedding of A into an ultrapower of B . More generally, the following is true for any class K of structures of the same signature Σ : 1. K is the class of all Σ -models of some set of first order sentences. 2. K is closed under elementary embeddings and ultraproducts. 3. K = EP U ( K ′ ) for some class of Σ -structures K ′ ( E denotes elementary embeddings and P U ultraproducts ) . Proof. ◮ (1) ⇒ (2) is immediate, as elementary embeddings and ultraproducts preserve all sentences ( n.b. closure under elementary embeddings implies closure under isomorphism ). ◮ (2) ⇒ (3) by taking K ′ to be K . ◮ Now assume (3) and consider Th ( K ′ ). If A | = Th ( K ′ ), then A ≡ B for some model B ∈ K ′ . By Theorem 7, we have A I / U ∼ = B I / U for some I and U . ◮ As A � A I / U , we have A � B I / U ; moreover B I / U ∈ P U ( K ′ ). ◮ So, A ∈ EP U ( K ′ ), proving (1). 10 / 31
Elementary classes characterised Theorem 7 A ≡ B if and only if there is an elementary embedding of A into an ultrapower of B . More generally, the following is true for any class K of structures of the same signature Σ : 1. K is the class of all Σ -models of some set of first order sentences. 2. K is closed under elementary embeddings and ultraproducts. 3. K = EP U ( K ′ ) for some class of Σ -structures K ′ ( E denotes elementary embeddings and P U ultraproducts ) . Proof. ◮ (1) ⇒ (2) is immediate, as elementary embeddings and ultraproducts preserve all sentences ( n.b. closure under elementary embeddings implies closure under isomorphism ). ◮ (2) ⇒ (3) by taking K ′ to be K . ◮ Now assume (3) and consider Th ( K ′ ). If A | = Th ( K ′ ), then A ≡ B for some model B ∈ K ′ . By Theorem 7, we have A I / U ∼ = B I / U for some I and U . ◮ As A � A I / U , we have A � B I / U ; moreover B I / U ∈ P U ( K ′ ). ◮ So, A ∈ EP U ( K ′ ), proving (1). 11 / 31
Digression: finite models Lemma 8 Let C be a class of finite models of some signature Σ . If C contains arbitrarily large finite models, then C is not elementary. Proof. ◮ Suppose C = Mod ( T ) for some theory T . ◮ Let ( A i : i ∈ ω ) be a sequence of models from C with strictly increasing sizes of their universes. Take A = � i ∈ ω A i / U , for a non-principal U . ◮ Then, A | = T , but A is infinite, so C ⊂ Mod ( T ), contradicting the assumption. Finite Model Theory Lemma 8 can be viewed as dashing all hope for a reasonable model theory of finite structures: completeness fails. Compactness also fails: say, the set { n < c : n ∈ N } (in the signature of natural numbers with an additional costant c ) is finitely satisfiable in the class of finite models, but has no finite model. Yet, Finite Model Theory is alive and well. This is mostly because Ehrenfeucht-Fra¨ ıss´ e games work. We will look at them later. 12 / 31
Digression: finite models Lemma 8 Let C be a class of finite models of some signature Σ . If C contains arbitrarily large finite models, then C is not elementary. Proof. ◮ Suppose C = Mod ( T ) for some theory T . ◮ Let ( A i : i ∈ ω ) be a sequence of models from C with strictly increasing sizes of their universes. Take A = � i ∈ ω A i / U , for a non-principal U . ◮ Then, A | = T , but A is infinite, so C ⊂ Mod ( T ), contradicting the assumption. Finite Model Theory Lemma 8 can be viewed as dashing all hope for a reasonable model theory of finite structures: completeness fails. Compactness also fails: say, the set { n < c : n ∈ N } (in the signature of natural numbers with an additional costant c ) is finitely satisfiable in the class of finite models, but has no finite model. Yet, Finite Model Theory is alive and well. This is mostly because Ehrenfeucht-Fra¨ ıss´ e games work. We will look at them later. 13 / 31
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