What is the theory ZFC without power set?

We show that the theory ZFC- , consisting of the usual axioms of ZFC but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC- in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of reals of every size ℵn , but none of size ℵω , and therefore, in which the collection axiom fails; there are models of ZFC- for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of ZFC- for which the Gaifman theorem fails, in that there is an embedding j:M→N of ZFC- models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings j:M→N of ZFC- models whose cofinal restriction j:M→⋃j""M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC- to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory ZFC− , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach .