Expressibility Example Let C=, + <,0, 1) and Th()be the full theory of N. there is MH Th() and a E M such that a is larger than every membe Let C*=LUcl, where c is a new constant symbol. We can construct a set of sentence 5={yn=1+1+…+1<c,n≥1} Then apply compactness theoremExpressibility . Example . . Let L = {·, +, <, 0, 1} and Th(N ) be the full theory of N . There is M |= Th(N ) and a ∈ M such that a is larger than every member. . Proof. . . Let L ∗ = L ∪ {c}, where c is a new constant symbol. We can construct a set of sentence S = {φn = 1 + 1 + · · · + 1 | {z } n < c, n ≥ 1}. Then apply compactness theorem. Yi Li (Fudan University) Discrete Mathematics June 9, 2013 10 / 15