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Is it possible for there to be a model of ZFC with the property that, for every set $S$ in the model, there is a unary predicate in the language of ZFC such that $S$ is the is the only set satisfying the predicate?

I'm pretty sure I've been told the answer is "yes", but I am never able to find a reference when I want one. So this question (assuming it is answered in the affirmative) is to be that reference!

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    $\begingroup$ I'm reminded of Joel David Hamkins's explanation of the subtleties of undefinable numbers here ... mathoverflow.net/questions/44102/… $\endgroup$
    – user4894
    Commented Dec 29, 2014 at 3:00
  • $\begingroup$ Was this actually a reference request? (maybe I should have asked first) $\endgroup$ Commented Dec 29, 2014 at 3:07
  • $\begingroup$ @user2345215: An external reference is fine. So is a self-contained answer. I mainly just want something I can point to to remind myself that yes, the answer really is "yes". And sometimes, to point other people to. $\endgroup$
    – user14972
    Commented Dec 29, 2014 at 3:11
  • $\begingroup$ Interesting question! Can one even find a model in which any class (not just the singletons) is definable? $\endgroup$
    – Hanno
    Commented Dec 29, 2014 at 19:17
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    $\begingroup$ @Hanno No, since any model of ZFC is infinite, and thus has at least continuum-many subsets, while there are only countably many formulas in the language of set theory. On the other hand, countable models have countably many elements, so it's possible that every element could be definable (which is what happens in pointwise definable models). $\endgroup$ Commented Apr 1, 2016 at 4:24

2 Answers 2

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Yes.

J. D. Hamkins, D. Linetsky, and J. Reitz, Pointwise definable models of set theory, Journal of Symbolic Logic 78(1), pp. 139-156, 2013.

Perhaps slightly surprisingly, every pointwise definable model of ZF satisfies the Axiom of Choice (because if every set is definable, then in particular it is ordinal definable, and V=HOD implies AC). Intuitively one might think to look to the AC to find a candidate for an undefinable set, but actually it is the opposite that is the case: If we're looking at a situation where AC fails, then there must be an undefinable set somewhere.

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The minimal model $M$ is such a model. $M$ is defined to be $L_{\delta}$ (the $\delta$-th level of constructible hierarchy) where $\delta$ is least such that $L_{\delta}$ models ZFC. Assuming that there are standard models of ZFC, the minimal model exists.

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    $\begingroup$ Do you have a reference for that, or are we suppose to take your word for it? :-) (This, if I recall correctly, is also discussed in the paper given by Henning in his answer.) $\endgroup$
    – Asaf Karagila
    Commented Dec 29, 2014 at 19:23
  • $\begingroup$ Use the definable well ordering of $M$ to conclude that the set $D$ of definable members of $M$ forms an elementary substructure of $M$. Now take the transitive collapse $D'$ of $D$ and argue, using minimality of $M$, that $D = D' = M$. $\endgroup$
    – user203787
    Commented Dec 29, 2014 at 19:32

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