In his *Set Theory. An Introduction to Indepencence Proofs*, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended interpretation of set theory he says such things as, for example, that the domain of discourse $V$ is the collection of all (well-founded, when foundation is introduced) hereditary sets.


This point of view has always made me feel a bit uncomfortable. How can a variable in a first-order language run over the elements of a collection that is not a set? Only recently I realized that one thing is to be a platonist, and another thing is to believe such an odd thing.


A first-order theory of sets with a countable language can only prove the existence of countably many sets. Let me call them provable sets for short. Platonistically, we wish our intended interpretation of that theory to be one in which every provable set is actually the set the theory says it is. So we don't need our interpretation to contain every set, we just need that it contains at least the true provable sets. This collection is, really, a set, although it doesn't know it.


To be a bit more concrete, if one is a platonist and the cumulative hierarchy is what one has in mind as the real universe of sets, one can think that the $V$ of one's theory actually refers to a an initial segment of that hierarchy, hence variables run no more over the real $V$ but only over the elements of some $V\_\alpha$.


There's a parallel to these ideas. For example, when we want to prove consistency with $ZFC$ of a given sentence, we do not directly look for a model of $ZFC$ where that sentence is true, but instead we take advantage of knowing that every finite fragment of $ZFC$ is consistent and that every proof involves only finitely many axioms.


My question is: then, is this position tenable or am I going awfully wrong? I apologize that this seems a philosophical issue rather than a mathematical one. I also apologize for stating things so simply (out of laziness).



Intended interpretations of set theories
While Kunen takes for universe the collection of all hereditary sets, Marc proposes to restrict the universe to those hereditary sets which are first-order definable without parameters (Marc's "provable sets"). To make this more mathematical, let me rephrase the question as follows:



> 
> Suppose $M$ is a model of ZF. Does the subset $K$ of $M$ consisting of the first-order definable (without parameters) elements of $M$ form a model of ZF?
> 
> 
> 


This turns out to be a delicate question which was answered [here on MO](https://mathoverflow.net/questions/10413/definable-collections-without-definable-members-in-zf) some time ago. To summarize the answer there, Marc's proposed universe $K$ will be a model of ZF if and only if $M$ is a model of ZF + [V = OD](http://en.wikipedia.org/wiki/Ordinal_definable_set).


Returning to the more philosophical question, this says that Marc's proposal is tenable if and only if there is a reason to believe that the hereditary sets are all ordinal definable. [Occam's Razor](http://en.wikipedia.org/wiki/Occam%39s_razor) type arguments support that V = OD and, indeed, V = L is true. However, since it is much weaker than the rigid assumption V = L, the assumption V = OD is not incompatible with a much richer view of the universe...




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There is another interpretation of Marc's "provable sets." Instead of merely requiring them to be first-order definable without parameters, one can require them to provably exist in any well-founded model of ZF. With this interpretation, one gets a potentially much smaller set: the minimal well-founded model of ZF. This minimal model is $L\_\alpha$ where $\alpha$ is the smallest element of $Ord\cup\{Ord\}$ such that $L\_\alpha \models ZF$. 


Note that it is entirely plausible that $\alpha = Ord$ (for example this will happen if we happen to live in the minimal model in question). The assumption that "$L\_\alpha$ is a set" is equivalent to the existence of a well-founded model of ZF. From a philosophical standpoint, this is a rather strong assertion. For example, it implies that ZF is not only consistent but also [$\omega$-consistent](http://en.wikipedia.org/wiki/%CE%A9-consistent_theory) and much more...


