 V. Gitman, J. D. Hamkins, and T. Johnstone, “What is the theory $\mathsf {ZFC}$ without power set?,” Mlq math. log. q., vol. 62, iss. 45, pp. 391406, 2016.
@ARTICLE{zfcminus:gitmanhamkinsjohnstone, AUTHOR = {Gitman, Victoria and Hamkins, Joel David and Johnstone, Thomas A.}, TITLE = {What is the theory {$\mathsf {ZFC}$} without power set?}, JOURNAL = {MLQ Math. Log. Q.}, FJOURNAL = {MLQ. Mathematical Logic Quarterly}, VOLUME = {62}, YEAR = {2016}, NUMBER = {45}, PAGES = {391406}, ISSN = {09425616}, MRCLASS = {03E30}, MRNUMBER = {3549557}, MRREVIEWER = {Arnold W. Miller}, DOI = {10.1002/malq.201500019}, URL = {http://dx.doi.org/10.1002/malq.201500019}, PDF={http://boolesrings.org/victoriagitman/files/2011/10/ZFC.pdf}, EPRINT={1110.2430}}
This is joint work with Joel David Hamkins and Thomas Johnstone.
Set theory without the power set axiom is used in arguments and constructions throughout the subject and is usually described simply as having all the axioms of $\rm{ZFC}$ except for the power set axiom. This theory arises frequently in the large cardinal theory of iterated ultrapowers, for example, and perhaps part of its attraction is an abundance of convenient natural models, including $\langle H_\kappa,{\in}\rangle$ for any uncountable regular cardinal $\kappa$, where $H_\kappa$ consists of sets with hereditary size less than $\kappa$. When prompted, many set theorists offer a precise list of axioms: extensionality, foundation, pairing, union, infinity, separation, replacement and choice. Let us denote by $\rm{ZFC}{}$ the theory having the axioms listed above with the axiom of choice taken to mean Zermelo’s wellordering principle, which then implies Zorn’s Lemma as well as the existence of choicefunctions. These alternative formulations of choice are not all equivalent without the power set axiom as is proved by Zarach in [1], in particular there are models of $\rm{ZF}^$ in which choicefunctions exist but Zermelo’s wellordering principle fails. Zarach initiated the program of establishing unintuitive consequences of set theory without power set, which we carry on in this article.
In this article, we shall prove that this formulation of set theory without the power set axiom is weaker than may be supposed and is inadequate to prove a number of basic facts that are often desired and applied in its context. Specifically, we shall prove that the following behavior can occur with $\rm{ZFC}{}$ models.
 (Zarach, [2]) There are models of $\rm{ZFC}{}$ in which the countable union of countable sets is not necessarily countable, indeed, in which $\omega_1$ is singular, and hence the collection axiom scheme fails.
 (Zarach [2]) There are models of $\rm{ZFC}{}$ in which every set of reals is countable, yet $\omega_1$ exists.
 There are models of $\rm{ZFC}{}$ in which for every $n<\omega$, there is a set of reals of size $\aleph_n$, but there is no set of reals of size $\aleph_\omega$.

The Łós ultrapower theorem can fail for $\rm{ZFC}{}$ models.
 There are models $M\models\rm{ZFC}{}$ with an $M$normal measure $\mu$ on a cardinal $\kappa$ in $M$, for which the ultrapower by $\mu$, using functions in $M$, is wellfounded, but the ultrapower map is not elementary.
 Such violations of Łós can arise even with internal ultrapowers on a measurable cardinal $\kappa$, where $P(\kappa)$ exists in $M$ and $\mu\in M$.

There is $M\models\rm{ZFC}{}$ in which $P(\omega)$ exists in $M$ and there are ultrafilters $\mu$ on $\omega$ in $M$,
but no such $M$ultrapower map is elementary.

The Gaifman theorem [3] can fail for $\rm{ZFC}{}$ models.
 There are $\Sigma_1$elementary cofinal maps $j:M\to N$ of transitive $\rm{ZFC}{}$ models, which are not elementary.
 There are elementary maps $j:M\to N$ of transitive $\rm{ZFC}{}$ models, such that the canonical cofinal restriction $j:M\to \bigcup j''M$ is not elementary.
 Seed theory arguments can fail for $\rm{ZFC}{}$ models. There are elementary embeddings $j:M\to N$ of transitive $\rm{ZFC}{}$ models and sets $S\subseteq\bigcup j'' M$ such that the seed hull $\mathbb X_S=\{j(f)(s)\mid s\in [S]^{{\lt}\omega}, f\in M\}$} of $S$ is not an elementary submodel of $N$.
 The collection of formulas that are provably equivalent in $\rm{ZFC}{}$ to a $\Sigma_1$formula or a $\Pi_1$formula is not closed under bounded quantification.
Nevertheless, these deficits of $\rm{ZFC}{}$ are completely repaired by strengthening it to the theory $\rm{ZFC}^$, obtained by using collection rather than replacement in the axiomatization above.
[Bibtex]
@incollection {zarach:unions_of_zfminus_models,
AUTHOR = {Zarach, Andrzej},
TITLE = {Unions of {${\rm ZF}^{}$}models which are themselves
{${\rm ZF}^{}$}models},
BOOKTITLE = {Logic {C}olloquium '80 ({P}rague, 1980)},
SERIES = {Stud. Logic Foundations Math.},
VOLUME = {108},
PAGES = {315342},
PUBLISHER = {NorthHolland},
ADDRESS = {Amsterdam},
YEAR = {1982},
MRCLASS = {03C62 (03E35)},
MRNUMBER = {673801 (84h:03086)},
MRREVIEWER = {M. Dubiel},
}
[Bibtex]
@incollection {Zarach1996:ReplacmentDoesNotImplyCollection,
AUTHOR = {Zarach, Andrzej M.},
TITLE = {Replacement {$\nrightarrow$} collection},
BOOKTITLE = {G\"odel '96 ({B}rno, 1996)},
SERIES = {Lecture Notes Logic},
VOLUME = {6},
PAGES = {307322},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1996},
MRCLASS = {03E30 (03E35)},
MRNUMBER = {1441120 (98g:03120)},
}
[Bibtex]
@incollection {gaifman:ultrapowers,
AUTHOR = {Gaifman, Haim},
TITLE = {Elementary embeddings of models of settheory and certain
subtheories},
BOOKTITLE = {Axiomatic set theory ({P}roc. {S}ympos. {P}ure {M}ath., {V}ol.
{XIII}, {P}art {II}, {U}niv. {C}alifornia, {L}os {A}ngeles,
{C}alif., 1967)},
PAGES = {33101},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence R.I.},
YEAR = {1974},
MRCLASS = {02K15 (02H13)},
MRNUMBER = {0376347 (51 \#12523)},
MRREVIEWER = {L. Bukovsky},
}
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Love your work! I’ve been thinking about ZF + Zorn’s + “All sets are countable”. Is there something I should watch out for?
Thanks, Christopher! Are you working in set theory? You might also be interested in http://boolesrings.org/victoriagitman/2013/06/25/ongroundmodeldefinability/.
I’m a student of set theory. I’m just getting into forcing, so the linked article is a little beyond me, but I’m sure I’ll get through it.
My favorite resource for beginner forcing is: http://www.math.ucsd.edu/~sbuss/CourseWeb/Math260_2012F2013W/Shoenfield_UnramifiedForcing.pdf
Good luck!
Christopher,
One thing to watch out for—and this is the main point of the paper mentioned here on Vika’s post—is that you have to be careful what you mean by ZF+Zorn. For example, if you mean to axiomatize this theory using the replacement axiom version of ZF, then you won’t be able to prove the collection axiom in your theory, and it could be that for every natural number $n$ you have witnesses $x$ such that $\varphi(n,x)$, but there is no single set $X$ containing such witnesses for every $n$. In the sense, the “right” version of your theory will be axiomatized with the collection and separation axioms, rather than merely with replacement.
One minor point is that one weakens the set theory, equivalents of the axiom of choice may no longer be equivalent. For example $V_{\omega+\omega}$ is a model of Zermelo’s set theory in which Zorn’s lemma holds (assuming choice holds in the universe, anyway), but not every set is isomorphic to an ordinal. However you can still find a wellordering on every set (it’s just that the order type is not an ordinal anymore).
In here one should note that once the assumption that every set is countable has been added, the axiom of choice is provable outright. Zorn’s lemma too. In fact the proof no longer requires any transfinite recursion. Countability ensures everything works out just fine as it is.
I thought so, but I wasn’t entirely sure. That’s good to know. Thank you guys.