Viva la difference I: Nonisomorphism of ultrapowers of countable models

Viva la difference II. The Ax-Kochen isomorphism theorem

We show in section 1 that the Ax-Kochen isomorphism theorem requires the continuum hypothesis. Most of the applications of this theorem are insensitive to set theoretic considerations. (A probable exception is the work of Moloney.) In section 2 we give an unrelated result on cuts in models of Peano arithmetic which answers a question on the ideal structure of countable ultraproducts of Z. In section 1 we also answer a question of Keisler and Schmerl regarding Scott complete ultrapowers of R .