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I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed? If not, are there other axiomatic set theories able to express them? As for Dedekind cuts, from my understanding (maybe wrong), any irrational in question must have a 1-1 explicit function to a natural number, in order for the method to work (As an example the square root of 2 has such a function).
It boils down to this: is there an isomorphism between the the set of irrationals (as "just" numbers) and a a set of pure sets?