TITLE: Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?
QUESTION [21 upvotes]: Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?

REPLY [5 votes]: The notion of constructive proof in mathematics, as understood by constructivists themselves, is an inherently informal notion. There are certainly many formalized analogues of constructive mathematics, which are worth studying. But constructivists reject the notion that any of these captures the notion of constructive proof, and they often make the stronger claim that constructive proof cannot be formalized.