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6 Lecture 10.10.05 LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 9 6 Lecture 10.10.05 These memoirs begin with material presented during lecture 5, but omitted from the memoir of that lecture. We observed that the notion of a strict linear order can be characterized by a first-order sentence, the conjunction of the following conditions: • (8x):x < x (irreflexivity) • (8x)(8y)(8z)(x < y ! (y < z ! x < z)) (transitivity) • (8x)(8y)(x 6= y ! (x < y _ y < x)) (comparability) We noted that for every natural number n there is a unique, up to isomor- phism, linear order on the set [n] = f1; : : : ; ng. On the other hand, Exercise 1 Show that there are 2@0 linear orders, up to isomorphism, on the set f1; 2; 3;:::g. 1 We gave several examples of infinite linear orders: 1. hN; <i (the natural numbers with their usual order); 2. hZ; <i (the integers with their usual order); 3. hQ; <i (the rational numbers with their usual order); 4. hR; <i (the real numbers with their usual order); 5. hN; ≺i, where i ≺ j if and only if i is even and j is odd, or the parity of i and j is the same and i < j, (the natural numbers with an unusual order). 2 We examined first-order conditions that distinguish among these orderings. 1. (9x)(8y):y < x (there is a least element) 2. (9x)(8y):x < y (there is a greatest element) 3. (8x)((9w)x < w ! (9y)(x < y ^ (8z):(x < z ^ z < y))) (discrete) 4. (8x)(8y)(x < y ! (9z)(x < z ^ z < y)) (dense) 5. (9x)((9y)y < x ^ (8z)(z < x ! (9w)(z < w ^ w < x))) (there is a limit point) Observe that orders (1)1 and 2 are discrete and have no limit point, moreover, the first of these has a least element while the second does not. The orders (1)3, 4, and 5 all have at least one limit point, moreover, the first two are dense while the last is discrete. Exercise 2 Show that hQ; <i ≡ hR; <i: LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 10 Here begins the memoir of lecture 6. Thus far we have been exploring the expressive power of first-order logic by looking at classes of structures which are first-order definable. We say a class of structures K is an elementary class if and only if there is a first-sentence ' such that K = Mod(') and we say K is and extended elementary class if and only if there is a set of first-order sentences Σ such that K = Mod(Σ). Another approach to the study of expressive power is via consideration of the relations which are first-definable on a fixed structure. If '(x1; : : : ; xn) is a formula with free variables among x1; : : : ; xn, and A is a structure which interprets all the non-logical symbols occurring in ', then '[A] denotes the n-ary relation defined by ' on A, that is, '[A] = f< a1; : : : ; an >j A j= '[a1; : : : ; an]g: We presented solutions to problems 2.2.9 and 2.2.11 in Enderton, which deal with definable collections of structures and definability within a fixed structure respectively. Let A = hjAj;P Ai be a structure for a language with a binary relation P and with no further relation, function, or constant symbols other than identity. We will often use A, rather than jAj, to denote the universe of A when no confusion is likely to result. If f is a function with domain A and range contained in A, that is, a function from A into A, we say that P A is the graph of f if and only if for all a; b 2 A; ha; bi 2 P A () f(a) = b: Let α be the sentence 8x9yP xy ^ 8x8y8z((P xy ^ P xz) ! y = z): Note that Mod(α) is the collection of all structures A such that P A is the graph of a function from A into A. Let β be the sentence 8x8y8z((P xz ^ P yz) ! x = y): Note that Mod(α ^ β) is the collection of all structures A such that P A is the graph of an injection (that is, 1-1 function) from A into A. Let γ be the sentence 8x9yP yx: Note that Mod(α ^ γ) is the collection of all structures A such that P A is the graph of a surjection from A onto A. Finally, note that Mod(α ^ β ^ γ) is the collection of all structures A such that P A is the graph of a permutation of A, that is, a bijection from A onto A. We next considered definability within the fixed structure N = hN; +; ·i where N = f0; 1; 2;:::g and + and · are the usual arithmetic operations on N: We considered the definability of simple sets and relations on N per exercise 2.2.11. (a) 8y(x + y = y)[N] = f0g: LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 11 (b) 8y(x · y = y)[N] = f1g: (c) 9z(8w(z · w = w) ^ x + z = y)[N] = fhm; ni j n = m + 1g: (d) 9z(8w(z + w 6= w) ^ x + z = y)[N] = fhm; ni j m < ng: We next considered the structures N = hN; <i and Z = hZ; <i where Z = f::: − 1; 0; 1;:::g: We noted that 8y:(y < x)[N] = f0g and asked whether f0g is definable in Z: The general sentiment was negative, but we agreed that we'd need a new idea to settle the question. To this end, we introduced the notion of an isomorphism of one structure onto another and of an automorphism, that is, an isomorphism of a structure onto itself. We say that f is an isomorphism of A onto B if and only if f is a bijection of A onto B and for all a; b 2 A ha; bi 2 P A () hf(a); f(b)i 2 P B: An automorphism of A is an isomorphism of A onto A. We write Aut(A) for the set of automorphisms of A. We stated the following theorem, which says that first-order logic satisfies a natural desideratum for a language to be logical { it does not distinguish between structurally identical models. Theorem 1 (Isomorphism Theorem) Suppose f is an isomorphism of A onto B. Then for every formula α(x1; : : : ; xn), with at most the variables indi- cated free, and for all a1; : : : ; an 2 A, A j= α[(x1ja1);:::; (xnjan)] () B j= α[(x1jf(a1));:::; (xnjf(an))]: As a corollary to the Isomorphism Theorem, we have the Corollary 1 (Automorphism Theorem) If f is an automorphism of A and α(x) is a formula with at most x free, then for all a 2 A, a 2 α[A] () f(a) 2 α[A]: We applied the automorphism theorem to show that the only sets definable in Z = hZ; <i are Z and ;: This follows from the observation that for every p 2 Z the function fp 2 Aut(Z), where fp(q) = p + q; for all q 2 Z: We continued to go over exercises from Enderton, Section 2.2. Exercise 2.2.15 provided another opportunity to apply the Automorphism Theorem to establish an undefinability result. We showed that the addition relation, that is, fhp; q; ri j p + q = rg, is not definable in the structure N = hN; ·i: We first observed that the ordering relation m < n is definable in terms of addition by the formula 9z(z + z 6= z ^ x + z = y): Therefore, it suffices to show that the ordering relation is not definable in N = hN; ·i: For this, it suffices to exhibit an automorphism h of N = hN; ·i which is not order preserving, that is, for some p; q we have p < q but h(q) ≤ h(p): We showed that there is an automorphism h of N = hN; ·i such that h(2) = 3 and h(3) = 2, thereby completing the exercise. We observed that any permutation of the prime numbers can be extended to an automorphism of N = hN; ·i: We concluded that there are uncountably many automorphisms of N = hN; ·i: LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein 12 Exercise 3 1. Let B = ff j f is a bijection from N onto Ng and let C = ff j f : N 7! f0; 1gg: Show that there is a bijection of B onto C. 2. With C as above show that there is a bijection from C onto Aut(hQ; <i). 7 Lecture 10.10.07 We continued our study of the expressive power of first-order logic. Today we focused on finite structures. We recalled our earlier observation that for every finite directed graph A, there is a sentence such that for all graphs directed graphs B, B j= () A =∼ B: Exercise 4 Let A be a finite structure that interprets an infinite collection of relation symbols Rn where Rn is n-ary. Show that for every structure B, B j= Th(A) () A =∼ B: With Exercise 2.2.16 we meet another measure of expressive power that is specially suited to measure expressiveness in the context of finite structures. Given a first-order sentence α, we define the spectrum of α as follows: Spec(α) = fn 2 N j 9A(card(A) = n and A j= αg: We are asked to exhibit a sentence α whose spectrum is the set of positive even numbers.
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