What is the theory ZFC without power set?

  • 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.

See Victoria Gitman’s summary post on the article

What is the theory of ZFC-Powerset? Toronto 2011

This was a talk at the Toronto Set Theory Seminar held April 22, 2011 at the Fields Institute in Toronto.

The theory ZFC-, consisting of the usual axioms of ZFC but with the powerset axiom removed, when axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the axiom of choice, is weaker than commonly supposed, and suffices to prove neither that a countable union of countable sets is countable, nor that $\omega_1$ is regular, nor that the Los theorem holds for ultrapowers, even for well-founded ultrapowers on a measurable cardinal, nor that the Gaifman theorem holds, that is, that every $\Sigma_1$-elementary cofinal embedding $j:M\to N$ between models of the theory is fully elementary, nor that $\Sigma_n$ sets are closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using the collection axiom rather than replacement in the axiomatization above. These results extend prior work of Zarach. This is joint work with Victoria Gitman and Thomas Johnstone.

Article | Victoria Gitman’s post