What is the theory without power set?

Abstract

We show that the theory , consisting of the usual axioms of 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 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 , but none of size , and therefore, in which the collection axiom fails; there are models of for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach .