Omega_1 Categorical Theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:23:29Z http://mathoverflow.net/feeds/question/60321 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60321/omega-1-categorical-theory Omega_1 Categorical Theory Eran 2011-04-01T21:07:42Z 2011-04-01T21:18:04Z <p>Hi,</p> <p>In Chang &amp; Keisler "Model Theory" it is claimed that the theory of a one-to-one function of A onto A with no finite cycles is $\omega_1$- categorical (page 140). Why is that, and is there a reference for this?</p> http://mathoverflow.net/questions/60321/omega-1-categorical-theory/60323#60323 Answer by Richard Dore for Omega_1 Categorical Theory Richard Dore 2011-04-01T21:18:04Z 2011-04-01T21:18:04Z <p>Let $A$ and $B$ be two models of size $\omega_1$. In both A and B, being in the same cycle is an equivalence relation. Each equivalence class has size $\omega$. So there are $\omega_1$ many equivalence classes in both A and B. Fix a bijection between the equivalence classes in A and B. You can use this to make an isomorphism between A and B because every single equivalence class in either A and B is isomorphic to every other.</p>