TITLE: Is it possible that every set can be specified?
QUESTION [16 upvotes]: 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!

REPLY [4 votes]: 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.