Hi,
In Chang & Keisler "Model Theory" it is claimed that the theory of a onetoone 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?
Hi, In Chang & Keisler "Model Theory" it is claimed that the theory of a onetoone 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? 


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. 

