Compactness Theorem satisfiable in arbitrarily large finite models is satisfiable in some<'s Let l be a first-order language. Any set S of sentences of l that infinite model Sketch Idea Suppose s is satisfiable in arbitrary large finite models. Let r be a 2-ary relation symbol that is not part of the language L, and enlarge L to L' by adding R We can modify the interpretation of R without affecting the truth values of members of s, sincer does not occur in members of S. so we can write a sentence An that asserts there are at least n thing We can imply Theorem by applying Compactness TheoremCompactness Theorem Let L be a first-order language. Any set S of sentences of L that is satisfiable in arbitrarily large finite models is satisfiable in some infinite model. Sketch Idea: Suppose S is satisfiable in arbitrary large finite models. Let R be a 2-ary relation symbol that is not part of the language L, and enlarge L to L 0 by adding R. We can modify the interpretation of R without affecting the truth values of members of S, since R does not occur in members of S. So we can write a sentence An that asserts there are at least n thing. We can imply Theorem by applying Compactness Theorem. Yi Li (Fudan University) Discrete Mathematics June 20, 2012 10 / 15