On a Theorem of Zarach Concerning the Global Pattern of Cardinals in Generic Extensions of Models of ZFC

Let M ZFC and let F: be a function on the ordinals of M.For which such F will there exist a model N of ZFC with thesame ordinals as M such that N is a cardinal if and only ifM () ( = _F())? A partial answer to this question was given in a theorem ofZarach. We present here a counter-example and correction tothis theorem. Our argument invokes results of Drake concerningthe collapse of weak cardinal powers in certain generic extensions. This work is supported in part by the National Research Councilof Canada under grant number A8216.