What is the theory ZFC without power set? (Q2827952)

The authors propose that ZFC without power set should not be identified with the theory ZFC\(^-\) obtained by removing the power set axiom from the usual axiomatization of ZF supplemented with the well-ordering principle. The standard for making this judgement is the theory of ultrapowers: A model of ZFC\(^-\) may not satisfy fundamental theorems of ultrapowers, as it is proved in the paper under review. The problem of ZFC\(^-\) is the weakness of the replacement scheme in the absence of the power set axiom, and this can be repaired by the use of the collection scheme instead of replacement. The preferred theory with collection scheme and without power set is denoted by ZFC\(^-\).