- V. Gitman, J. D. Hamkins, and T. A.~Johnstone, “What is the theory ZFC without Powerset?,” Mathematical Logic Quarterly, vol. 62, iss. 4–5, pp. 391-406, 2016.
`@ARTICLE{GitmanHamkinsJohnstone:WhatIsTheTheoryZFC-Powerset?, AUTHOR = {Victoria Gitman and Joel David Hamkins and Thomas A.~Johnstone}, TITLE = {What is the theory {ZFC} without {Powerset}?}, JOURNAL = {Mathematical Logic Quarterly}, YEAR = {2016}, volume = {62}, number = {4--5}, pages = {391--406}, month = {}, note = {}, abstract = {}, keywords = {}, doi = {10.1002/malq.201500019}, eprint = {1110.2430}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://arxiv.org/abs/1110.2430}, source = {}, ISSN = {0942-5616}, MRCLASS = {03E30}, MRNUMBER = {3549557}, MRREVIEWER = {Arnold W. Miller}, URL = {http://dx.doi.org/10.1002/malq.201500019}, }`

This is joint work with Victoria Gitman and Thomas Johnstone.

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 $\omega_1$ is singular, in which every set of reals is countable, yet $\omega_1$ exists, in which there are sets of reals of every size $\aleph_n$, but none of size $\aleph_\omega$, and therefore, in which the collection axiom sceme fails; there are models of ZFC- for which the Los 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 $j:M\to N$ of ZFC- models that is $\Sigma_1$-elementary and cofinal, but not elementary; there are elementary embeddings $j:M\to N$ of ZFC- models whose cofinal restriction $j:M\to \bigcup j“M$ is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC- to a $\Sigma_1$-formula or a $\Pi_1$-formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory $\text{ZFC}^-$, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.

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Is it correct to say that Replacement is a consequence of $ZFC^{-}$?

Yes, that is correct. The point is that collection + separation are a strengthening of replacement, and the usual way of describing ZFC- in terms of replacement is too weak, not that it is too strong. The right axiomatization uses collection+separation, which implies replacement (so you could also describe it as collection+replacement).

Thanks. Very helpful.

By the way , is there an axiom (call it A) or set of axioms (call them $\mathscr A$) which, when added to $ZFC$-Powerset+Collection +Separation, will derive Powerset as a theorem (other than, of course, Powerset itself)?

I saw that you had asked this on MO, but I don’t have any useful nontrivial answer.

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