Logic: Lecture 13, 15 September 2015 ------------------------------------ Recall: Model theory X a set of FO sentences. Mod(X) = { M | M |= X } Elementary and Delta-Elementary classes of structures C is elementary if C = Mod(phi) for some FO sentence phi. C is Delta-elementary if C = Mod(X) for some set of FO sentences X. Examples: 1. A torsion group is one in which every element is of finite order Ax (x = 0 or x + x = 0 or ....) Not Delta-elementary Suppose X is a set of sentences characterizing torsion. Let Y = X U { ~(x + x + ... + x = 0) | n >= 1 } ------------------- n-times Every finite subset Y0 of Y is satisfiable. The second set is bounded by some n0. Choose a cyclic group of order > n0 and assign x some generator of the group. Hence, Y is satisfiable in (M,sigma). But sigma(x) cannot have finite order, so the underlying group is torsion free. 2. Connected graphs are not Delta-elementary Suppose X characterizes connectedness Let psi_n = ~(x = y) and ~Ex1..Exn( x1 = x and xn = y and E(x1,x2) and ... and E(xn-1,xn) ) Let Y = X U { psi_n | n >= 2} Every finite subset Y0 is satisfiable. There is a bound n0 for psi_n. Choose a cyclic polygon with 2n0 vertices. Y is satisfiable in (M,sigma) but sigma(x) and sigma(y) are not connected. Elementary equivalence M == M' wrt L if, for every sentence phi in L, M |= phi iff M' |= phi Th(M) = { phi | M |= phi) Lemma M == M' iff M' |= Th(M) Observation If two structures are isomorphic, they are elementary equivalent. Theorem For any structure M, C = { M' | M' == M } is Delta-elementary, given by Mod(Th(M)). C is the smallest Delta-elementary elementary class containing M. Proof If M in Mod(X), then M |= X, so M' == M also satisfies M' |= X so M' in Mod(x). So C subset of Mod(X). Theorem If M is infinite, the class of structures isomorphic to M is not Delta-elementary. Proof By Upward Lowenheim-Skolem theorem. Can find a model of size powerset of M. Categorical theory: all models are isomorphic. No FO theory can be categorical (by upward and downward LS theorems) Peano's axioms for (N,s,0) 1. 0 is not the successor of any element 2. successor is injective 3. For all X, if 0 in X and n in X implies s(n) in X, then X = N Second order axioms. This is categorical in second order logic but not in FO. Nonstandard model of arithmetic: Structure that is elementary equiv but not isomorphic to (N,s,0) satisfying Th(N) --- all FO sentences that hold by virtue of Peano's axioms. Theorem (Skolem) There exists a countable nonstandard model of arithmetic X = Th(N) U { ~(x = 0), ~(x = 1) .... } Again, by compactness, there is a (countable) model M' for X in which sigma(x) not in N. What does the model look like? sigma(x) = a, but a is outside N a must be closed under predecessor and successor (Th(N)) ---------- --------------------------- 0 1 2 ... ... a-2 a-1 a a+1 a+2 ... a+a must give us another copy of the same to the right (if a+a is in the second copy, then a+a = a+n for some n, but then a = n by cancellation) By midpoint theorem (Am An Ep m+n = 2p or m+n = 2p + 1). Midpoint of a in one copy and b in another copy must be in a different copy in between (otherwise a and b would lie in the same copy) So, Skolem's theorem implies there is a model of Th(N) that looks like Q with each rational replaced by a copy of N ======================================================================