Characterizing First Order Logic

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1 Characterizing First Order Logic Jared Holshouser, Originally by Lindstrom September 16, 2014 We are following the presentation of Chang and Keisler. 1 A Brief Review of First Order Logic Definition 1. A language L is a collection of symbols, which are broken into three groups: relation symbols, function symbols and constant symbols. Definition 2. Let L be a language. A model for L is a pair A = (A, I), where we call A the universe of A and I an interpretation function. I must be as follows: Each n-ary function symbol F gets taken to an n-ary function G : A n A, Each n-ary relation symbols R gets taken to an n-ary relation P A n, and Each constant symbol c gets taken to a constant x A. Proposition 1. Suppose A is a model for the language L. Let X be a set. Then we can expand A to a model of L = A X. Proof. WLOG X is disjoint from L. Let I be an interpretation on X. Then A = (A, I I ) suffices. Definition 3. For A, A, L, L, and X as above, we call A the expansion of A to L and A the reduct of A to L. Definition 4. Two models A = (A, I) and B = (B, J ) of L are isomorphic if there is a function f : A B so that f is a bijection, For each relation symbol R, if R is n-ary, then for all x 1,, x n A I(R)(x 1,, x n ) J (R)(f(x 1 ),, f(x n )), For each function symbol F, if F is n-ary, then for all x 1,, x n A For each constant symbol c, f(i(c)) = J (c). We write f : A = B. I(F )(x 1,, x n ) = J (F )(f(x 1 ),, f(x n )), 1

2 Definition 5. Suppose A = (A, I) and B = (B, J ) are models of L. Then B is a submodel of A if B A and the following hold: for R an n-ary relation symbol, J (R) = I(R) B n, for F an n-ary function symbol, J (F ) = I(f) B n, and for c a constant symbol, J (c) = I(c). We now define first order logic, recursively. First the allowable symbols: We have parentheses (,); variables v 1,, v n, ; connectives and ; and the quantifier. Note that these are used in addition to the symbols of L and the symbol =. Definition 6. The collection of terms is defined recursively as follows: 1. A variable is a term, 2. a constant symbol is a term, and 3. if F is an n-ary function symbol, and t 1,, t n are terms, then F (t 1,, t n ) is a term. Definition 7. The collection of atomic formulas is defined recursively as follows: 1. if t 1 and t 2 are terms, then t 1 = t 2 is an atomic formula, 2. if R is an n-ary relation symbol and t 1,, t n are terms, then R(t 1,, t n ) is an atomic formula. Definition 8. The collection of formulas is defined recursively as follows: 1. atomic formulas are formulas, 2. if φ and ψ are formulas, then φ ψ and φ are formulas, 3. if v is a variable, and φ is a formula, then vφ is a formula. Definition 9. A sentence is a formula with no free variables. We now define model satisfaction: Definition 10. Fix a model A of a language L. We will define A = φ[ x] for formulas φ, and t[ x] for terms t. We begin with terms. If t is v i, then t[ x] := x i. If t is c, then t[ x] := I(c). If t is F (t 1,, t n ), then t[ x] = I(F )(t 1 [ x],, t n [ x]). Now we do atomic formulas. Suppose φ is t 1 = t 2. Then A = φ( x) t 1 [ x] = t 2 [ x]. Suppose φ is R(t 1,, t n ). Then A = φ( x) I(R)(t 1 [ x],, t n [ x]). Finally we do formulas. A = (φ ψ) iff A = φ and A = ψ. A = φ iff it is false that A = φ. A = v i φ[x 1,, x m ] (for i m) iff for all a A, A = φ[x 0,, x i 1, a, x i+1,, x m ] Remark 1. For sentences φ, we can write A = φ instead of A = φ[ x], as it does not depend on x. 2

3 Definition 11. Two models A and B of a language L are elementarily equivalent iff whenever φ is a sentence we have that A = φ iff B = φ. We write A B. Proposition 2. If A = B, then A B. Definition 12. Let Σ be a set of sentences of L. Then Σ has a model if there is a model A of L so that A = φ for all φ Σ. Theorem 1 (Compactness Theorem). A set of sentences Σ has a model iff every finite subset of Σ has a model. Theorem 2 (Downward Lowehnheim-Skolem Theorem). Every every consistent set of sentences in the language L has a model of size max{ω, L }. In particular, every valid sentence has a countable model. 3

4 2 Abstract Logics From here on, in order to simplify the argument, we restrict the discussion to lanaguages without function symbols. Definition 13. An abstract logic is a pair of classes (l, = l ) with the following properties: l is the class of sentences and = l is the satisfaction relation of the logic (l, = l ). 1. The Occurrence Property: For each φ l there is associated a finite language L φ called the set of symbols occuring in φ. The relation A = l φ is a relation between sentences φ of l and models A for languages L which contain L φ. 2. The Expansion Property: The relation A = l φ depends only on the reduct of A to L φ. I.e. if A = l φ and B is an expansion of A to a larger language, then B = l φ. 3. The Isomorphism Property: The relation A = l φ is preserved under isomorphism. 4. The Renaming Property: The relation A = l φ is preserved under renaming. That is, if we relabel the symbols of the logic, and modify the model accordingly, = l is preserved. 5. The Closure Property: l contains all atomic sentences, l is closed under the usual first order connectives, = l satisfies the usual rules for satisfaction of atomic formulas and first order connectives, and the set of symbols L φ behaves as expected for atomic sentences and first order connectives. 6. The Quantifier Property: l is closed under universal and existential quantifiers, and behaves as expected for them. 7. The Relativization Property: For each sentence φ l and relation R(x, b 1,, b n ) with R, b 1,, b n not in L φ, there is a new sentence φ R(x, b 1,, b n ), read φ relativized to R(x, b 1,, b n ) which has the set of symbols L φ {R, b 1,, b n } and is such that whenever B is the submodel of a model A for L φ with universe B = {a A : R(a, b 1,, b n )}, we have (A, R, b 1,, b n ) = l φ R(x, b 1,, b n ) iff B = l φ Remark 2. This notion of abstract logic does not have free variables, so l should be considered as a collection of sentences. Proposition 3. First order logic is an abstract logic. Proof. Properties 1-6 are either standard theorems or obviously true. We will prove that property 7 holds. Let φ be a sentence of first order logic, and R(x, b 1,, b n ) be a relation. Then the relativization of φ is formed as follows. Replace the quantifier xψ by and replace the quantifier xψ by x[r(x, b 1,, b n ) = ψ] x[r(x, b 1,, b n ) ψ] Remark 3. Every abstract logic l contains first order logic by the closure and quantifier properties. Definition 14. Two abstract logics l, l are equivalent if for all φ l, there is a ψ l so that L φ = L ψ and A = l φ iff A = l ψ for all A; and similarly for all ψ l. 4

5 Definition 15. By a model of a set T of sentences of an abstract logic l we mean a model A so that A = l φ for all φ T. Two models A and B for the same language l are l-elementarily equivalent if for each sentence φ l, A = l φ iff B = l φ. Definition 16. An abstract logic l is countably compact iff for every countable set T l, if every finite subset of T has a model, then T has a model. The Lowenheim number of l is the least cardinal α such that every sentence of l which has a model, has a model of power at most α. Remark 4. If the Lowenheim number of a logic l exists, it must be at least ω, as l contains every sentence of first order logic. Proposition 4. Let l be an abstract logic such that for each finite language L, the class {φ l : L φ L} is a set. Then the Lowenheim number of l exists. Proof. By renaming, we can WLOG assume that if φ l, then L φ V ω. This makes l a set. Define a map α : l ON by { ω if φ has no models α(φ) = κ if κ is the size of the smallest model of φ Then sup{α(φ) : φ l} exists, and is the Lowenheim number of l. Definition 17. Let A and B be models for a language L. A partial isomorphism I : A = p B is a relation I on the set of pairs of finite sequences (a 1,, a n ), (b 1,, b n ) of elements of A and B of the same length such that 1. I 2. If (a 1,, a n )I(b 1,, b n ), then (A, a 1,, a n ) and (B, b 1,, b n ) satisfy the same atomic sentences of L(c 1,, c n ) 3. If (a 1,, a n )I(b 1,, b n ), then and vice versa. c A d B[(a 1,, a n, c)i(b 1,, b n, d)] Proposition 5. Any two finite or countable partially isomorphic models are isomorpic. Proposition 6. Let l be an abstract logic which has Lowenheim number ω. Then any two models which are partially isomorphic are l-elementarily equivalent. Proof. Suppose A and B are models for a language L and I : A = p B. Suppose by way of contradiction that there is a φ l so that A = l φ and B = l φ. WLOG we may assume that L = L φ and that I is preserved under subsequences. For the first, use reducts, and for the second extend I in the obvious way. Define F : A <ω A A <ω and F : A <ω A <ω A <ω by and F ((a 1,, a n ), b) = (a 1,, a n, b) F ((a 1,, a n ), (b 1,, b m )) = (a 1,, a n, b 1,, b m ) Let A = (A A <ω, A, F, F ) be a model, where F and F are treated as relations. Similarly build B = (B B <ω, B, G, G ). WLOG, by the isomorphism property, A,A <ω, B, and B <ω are all disjoint from each other. Now consider C = (A, B, I). Then by the closure, quantifier, and relativization properties of l, there is a sentence ψ l so that C = l ψ and ψ = [(A = l φ) (B = l φ) (A = p B)] Then ψ has a countable model C 0, which gives us models A 0 and B 0 for L. Then A 0 = B0. This contradicts the isomorphism property, as C 0 = ψ. 5

6 3 The Main Theorem Theorem 3 (Lindstrom s Characterization of First Order Logic). Suppose l is an abstract logic which is countably compact and has Lowenheim number ω. Then l is equivalent to first order logic. Proof. Let l be a countably compact abstract logic with Lowenheim number ω. Let φ be a sentence of first order logic, and A be a model of L φ. Then by the closure and quantifier properties, A = φ iff A = l φ. Now we must prove the other direction. We first define some relations between models of a finite language L. Let A and B be models of L. We define relations I k for k ω between A and B recursively. Let (a 1,, a n ) A n and (B 1,, b n ) B n. ai 0 b iff a and b satisfy the same atomic formulas. ai k+1 b iff 1. c A d B((a 1,, a n, c)i k (b 1,, b n, d)) and 2. d B c A((a 1,, a n, c)i k (b 1,, b n, d)). We say A k B if I k. Since L is finite and has no function symbols, for each k there is a finite set Γ k of first order logic in L such that for all models A and B of L, A k B iff A and B agree on Γ k. Let φ l be a sentence so that L φ L. It suffices to show that there is a k so that for all models A and B of L, (A k B and A = l φ) = B = l φ For then we will know that for some boolean combination of sentences in Γ k, ψ, L φ = L ψ and for all models A of L φ, A = l φ iff A = ψ. We proceed by way of contradiction. For each k choose models A k and B k so that A k k B k and A = l φ and B = l φ (1) Taking a subsequence if necessary, we can assume that the A k satisfy the same atomic sentences. By the isomorphism property, we can assume that the A k interpret the constants of L the same way. Let A = k A k. Each A k is a submodel of A, and we can take the A k so that A k ω =. As in the previous proof, let A = (A A <ω, A, F, F ) Similarly define B and B. Let where R, S and I are so that C = (A, B, R, S, ω,, I) A k = {a A : R(a, k)}, B k = {b B : S(b, k)}, A <ω k = { a A <ω : R( a, k)}, B k <ω = { b B <ω : S( b, k)}, and for each k, I( a, b, k) holds iff ai k b. By the closure, quantifier, and relativization properties there is a sentence θ l so that C = l θ and θ implies that (ω, ) is a total order with immediate successors and predecessors except for the first element, and (1) holds for all k ω. By countable compactness, θ has a model Ĉ = (Â, ˆB, ˆR, Ŝ, ˆω, ˆ, Î) so that ˆω has a nonstandard element H. Then ÂH = l φ, ˆBH = l φ, and ÂH H ˆBH. Define J between ÂH and ˆB H by (a 1,, a n )J(b 1,, b n ) iff (a 1,, a n )ÎH n(b 1,, b n ) Then J is a partial isomorphism between ÂH and ˆB H. But then by the previous proposition and the fact that l has Lowenheim number ω, ÂH l ˆBH. This is a contradiction. 6