There are different schools of constructivism. But if you insist on only dealing with mathematical objects that admit of finite descriptions with finite symbol

There are different schools of constructivism. But if you insist on only dealing with mathematical objects that admit of finite descriptions with finite symbols, there is a surjection from a countable set onto the entire universe of possible mathematical objects. That said, an enumeration of all mathematical objects is not possible. That's because we may not be able to resolve the question of whether 2 descriptions of a mathematical object actually refer to the same object or not.