Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I don't have a precise reference, but I am sure some people would call them "Quine corners". I think that they appear in the writings of Quine significantly before Shoenfield.
@MattF. Well, yes, but I want to stress the point, that given a model, ZFC is able to prove things about that very model. This would be either obscured or not expressed correctly, if I wanted to put all the statements with the existential statement in the precedent.
@PeterLeFanuLumsdaine Well that is true but it also partly answers the question. In ZFC having classes whose intersection does not exist means precisely this: I have a family of definable subsets of the universe, whose intersection is not definable.
If you don't \emph{heavily} change your foundations (i.e. you stick to classical logic and you want to have some very basic set theory or combinatorics), then no. You are not allowed to take arbitrary families of classes indexed over meta-theory..
