This question sounds weird to me.

I find it best not to speak about "existence", but speak instead of logical models that work. For example, we don't know if our concept of integers is consistent, but we have evolved a set of tools for reasoning about it that have been quite useful so far. Now we try to add new reasoning tools, new concepts, without breaking the system. For example, if we imagine "the set of all sets" and apply some common reasoning to it, we reach Russell's paradox; but we can't feed this paradox back into the integers to demonstrate their inconsistency, so we just throw the problematic concept away with no harm done.