Many-Sorted First-Order Model Theory Lecture 5 2 nd July, 2020 1 / - PowerPoint PPT Presentation

Many-Sorted First-Order Model Theory Lecture 5 2 nd July, 2020 1 / 56

A few comments on terminology and notation ◮ Our official terminology and notation is very useful for analysing properties of logics (note the plural!) in an abstract form. ◮ It is however quite cumbersome to use in practice, especially when the background logic is fixed. And it is non-standard! ◮ From now on I will use the standard terminology and notation. Officially, they will be treated as useful shorthands. ◮ Below is a dictionary to translate between the two. Our official Standard (our shorthand) sentence formula sentence with no variables sentence sentence in signature Σ[ x ] formula in signature Σ with free variables x expansion of A to signature Σ[ x ] valuation of x into A = Σ[ x ] ϕ with x A = a A | A | = ϕ ( a ) σ A (a function symbol) f A π A (a relation symbol) R A category theory common sense 2 / 56

A few comments on terminology and notation ◮ Our official terminology and notation is very useful for analysing properties of logics (note the plural!) in an abstract form. ◮ It is however quite cumbersome to use in practice, especially when the background logic is fixed. And it is non-standard! ◮ From now on I will use the standard terminology and notation. Officially, they will be treated as useful shorthands. ◮ Below is a dictionary to translate between the two. Our official Standard (our shorthand) sentence formula sentence with no variables sentence sentence in signature Σ[ x ] formula in signature Σ with free variables x expansion of A to signature Σ[ x ] valuation of x into A = Σ[ x ] ϕ with x A = a A | A | = ϕ ( a ) σ A (a function symbol) f A π A (a relation symbol) R A category theory common sense 3 / 56

A few comments on terminology and notation ◮ Our official terminology and notation is very useful for analysing properties of logics (note the plural!) in an abstract form. ◮ It is however quite cumbersome to use in practice, especially when the background logic is fixed. And it is non-standard! ◮ From now on I will use the standard terminology and notation. Officially, they will be treated as useful shorthands. ◮ Below is a dictionary to translate between the two. Our official Standard (our shorthand) sentence formula sentence with no variables sentence sentence in signature Σ[ x ] formula in signature Σ with free variables x expansion of A to signature Σ[ x ] valuation of x into A = Σ[ x ] ϕ with x A = a A | A | = ϕ ( a ) σ A (a function symbol) f A π A (a relation symbol) R A category theory common sense 4 / 56

A few comments on terminology and notation ◮ Our official terminology and notation is very useful for analysing properties of logics (note the plural!) in an abstract form. ◮ It is however quite cumbersome to use in practice, especially when the background logic is fixed. And it is non-standard! ◮ From now on I will use the standard terminology and notation. Officially, they will be treated as useful shorthands. ◮ Below is a dictionary to translate between the two. Our official Standard (our shorthand) sentence formula sentence with no variables sentence sentence in signature Σ[ x ] formula in signature Σ with free variables x expansion of A to signature Σ[ x ] valuation of x into A = Σ[ x ] ϕ with x A = a A | A | = ϕ ( a ) σ A (a function symbol) f A π A (a relation symbol) R A category theory common sense 5 / 56

Applications of compactness 6 / 56

Applications of compactness Theorem 1 (Compactness) A theory T has a model iff every finite subset of T has a model. Theorem 2 (It takes infinity to recognise infinity) Let ψ be a sentence in the pure equality language (no function symbols, no relation symbols except = ). If ψ holds in all infinite models, then there is an n ∈ N such that ψ holds in all models S with card ( S ) > n. Proof. ◮ Let T n = {¬ ψ, ¬ ϕ 1 , . . . , ¬ ϕ n } , where ϕ n are the sentences saying that there are precisely n elements. ( Exercise: write such sentences. ) ◮ Suppose each T n has a model S n . Then card ( S n ) ≥ n + 1. ◮ Put T = � n ∈ N T n . By compactness, T has a model, say, S . ◮ For each n ∈ N we have that S has strictly more elements than n , so S is infinite. ◮ Yet, S | = ¬ ψ . Contradiction. 7 / 56

Applications of compactness Theorem 1 (Compactness) A theory T has a model iff every finite subset of T has a model. Theorem 2 (It takes infinity to recognise infinity) Let ψ be a sentence in the pure equality language (no function symbols, no relation symbols except = ). If ψ holds in all infinite models, then there is an n ∈ N such that ψ holds in all models S with card ( S ) > n. Proof. ◮ Let T n = {¬ ψ, ¬ ϕ 1 , . . . , ¬ ϕ n } , where ϕ n are the sentences saying that there are precisely n elements. ( Exercise: write such sentences. ) ◮ Suppose each T n has a model S n . Then card ( S n ) ≥ n + 1. ◮ Put T = � n ∈ N T n . By compactness, T has a model, say, S . ◮ For each n ∈ N we have that S has strictly more elements than n , so S is infinite. ◮ Yet, S | = ¬ ψ . Contradiction. 8 / 56

Applications of compactness Theorem 1 (Compactness) A theory T has a model iff every finite subset of T has a model. Theorem 2 (It takes infinity to recognise infinity) Let ψ be a sentence in the pure equality language (no function symbols, no relation symbols except = ). If ψ holds in all infinite models, then there is an n ∈ N such that ψ holds in all models S with card ( S ) > n. Proof. ◮ Let T n = {¬ ψ, ¬ ϕ 1 , . . . , ¬ ϕ n } , where ϕ n are the sentences saying that there are precisely n elements. ( Exercise: write such sentences. ) ◮ Suppose each T n has a model S n . Then card ( S n ) ≥ n + 1. ◮ Put T = � n ∈ N T n . By compactness, T has a model, say, S . ◮ For each n ∈ N we have that S has strictly more elements than n , so S is infinite. ◮ Yet, S | = ¬ ψ . Contradiction. 9 / 56

Infinity is not finitely axiomatisable Definition 3 A class C of models is an elementary class, if C = Mod Φ for some set Φ of sentences. Then we say that C is axiomatised by Φ. If C is axiomatised by some finite Φ, then it is said to be finitely axiomatisable. Theorem 4 The class INF of infinite pure equality structures is not finitely axiomatisable. Proof. ◮ Suppose INF is axiomatised by a finite set Φ. ◮ Then ψ = � Φ has the properties from Theorem 2. ◮ Thus, ψ holds in some finite models. Contradiction. ◮ But INF is an elementary class. It is axiomatised by {¬ ϕ n : n ∈ N } . 10 / 56

Infinity is not finitely axiomatisable Definition 3 A class C of models is an elementary class, if C = Mod Φ for some set Φ of sentences. Then we say that C is axiomatised by Φ. If C is axiomatised by some finite Φ, then it is said to be finitely axiomatisable. Theorem 4 The class INF of infinite pure equality structures is not finitely axiomatisable. Proof. ◮ Suppose INF is axiomatised by a finite set Φ. ◮ Then ψ = � Φ has the properties from Theorem 2. ◮ Thus, ψ holds in some finite models. Contradiction. ◮ But INF is an elementary class. It is axiomatised by {¬ ϕ n : n ∈ N } . 11 / 56

Infinity is not finitely axiomatisable Definition 3 A class C of models is an elementary class, if C = Mod Φ for some set Φ of sentences. Then we say that C is axiomatised by Φ. If C is axiomatised by some finite Φ, then it is said to be finitely axiomatisable. Theorem 4 The class INF of infinite pure equality structures is not finitely axiomatisable. Proof. ◮ Suppose INF is axiomatised by a finite set Φ. ◮ Then ψ = � Φ has the properties from Theorem 2. ◮ Thus, ψ holds in some finite models. Contradiction. ◮ But INF is an elementary class. It is axiomatised by {¬ ϕ n : n ∈ N } . 12 / 56

Infinity is not finitely axiomatisable Definition 3 A class C of models is an elementary class, if C = Mod Φ for some set Φ of sentences. Then we say that C is axiomatised by Φ. If C is axiomatised by some finite Φ, then it is said to be finitely axiomatisable. Theorem 4 The class INF of infinite pure equality structures is not finitely axiomatisable. Proof. ◮ Suppose INF is axiomatised by a finite set Φ. ◮ Then ψ = � Φ has the properties from Theorem 2. ◮ Thus, ψ holds in some finite models. Contradiction. ◮ But INF is an elementary class. It is axiomatised by {¬ ϕ n : n ∈ N } . 13 / 56

Robinson’s principle Theorem 5 (Robinson’s principle) Let ϕ be a first-order sentence in the language of fields. If ϕ holds in all fields of characteristic 0 , then there is a prime p such that ϕ holds in all fields of characteristic ≥ p. Proof. ◮ Let Φ be some first-order rendering of field axioms, and for each prime p let χ p be the sentence 1 + 1 + · · · + 1 = 0. � �� � p times ◮ Let ∆ p = {¬ χ q : q ≤ p } (∆ p says: characteristic is greater than p ). ◮ Now, suppose there is an infinite sequence of primes ( p i ) i ∈ I , such that each set Σ i = {¬ ϕ } ∪ ∆ p i ∪ Φ, has a model. ◮ So, for each prime p there is a field of characteristic greater than p in which ϕ fails . ◮ By compactness, Σ = � i ∈ I Σ i has a model, say, K . ◮ Then, K is a field of characteristic 0, and K | = ¬ ϕ . ◮ But K | = ϕ by assumption. Contradiction. 14 / 56