  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 .
