MODELS OF SET THEORY: EXTENSIONS AND DEAD-ENDS

Abstract

This paper is a contribution to the study of extensions of arbitrary models of ZF (ZermeloFraenkel set theory), with no regard to countability or well-foundedness of the models involved. Our main results include the theorems below; in Theorems A and B, N is said to be a conservative elementary extension of M if N elementarily extends M, and the intersection of every N-definable set with the universe of M is M-definable (parameters allowed). In Theorem B, ZFC is the result of augmenting ZF with the axiom of choice.