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 ZFCin 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 ZFCfor which the Loś theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFCfor which the Gaifman theorem fails, in that there is an embedding j : M → N of ZFCmodels that is Σ1-elementary and cofinal, but not elementary; there are elementary embeddings j : M → N of ZFCmodels whose cofinal restriction j : M → ⋃ j "M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFCto a Σ1-formula or a Π1-formula is not closed under bounded quantification. Nevertheless, these deficits of ZFCare 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 [Zar96].