Do there exist theorems of the form: "For every set theory there always exist at least two sets $M$ and $N$ between which there exists bijections and at least one bijection of those bijections cannot be explicitly constructed in that set theory."?

What I mean is, approximately, the following: If we have some set theory then that set theory is determined very much with the set of axioms of that set theory and with its notation.

So this question is really about: Could always $M$ and $N$ and bijections between them be found in concrete set theories but that they are such that, no matter how sophisticated the notation of that concrete set theory is, at least one bijection between $M$ and $N$ cannot be constructed explicitly in that set theory?

My expertise in set theory is mostly at the undergraduate level, and I wouldn´t be much surprised if theories dealing with questions like this one are already well-developed and constructed.

So, if you know how to easily explain to me the issues of this question and results of the form mentioned, like you would explain it to an average undergraduate, that would be nice.