Power-Like Models of Set Theory

A model 9) = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be 0-like, where E interprets E and 0 is an uncountable cardinal, if I MI 0 but I{b E M: bEa}I < 0 for each a E M. An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an 1 I -like model. Coupled with Chang's two cardinal theorem this implies that if 0 is a regular cardinal 0 such that 2<' = 0 then every consistent extension of ZFalso has a 0+-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZFhas an 82-like model. Here we prove: THEOREM A. If 0 has the tree property then the following are equivalent for any completion T of ZFC: (i) T has a 0-like model. (ii) (D C T, where D is the recursive set of axioms {13n(I is n-Mahlo and " V,, is a En -elementary submodel of the universe"): n E co }. (iii) T has a A-like modelfor every uncountable cardinal A. THEOREM B. The following are equiconsistent over ZFC: (i) "There exists an co-Mahlo cardinal". (ii) "For everyfinite language i, all 82-like models of ZFC(Y) satisfy the scheme F(Y). ?