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It isn't so simple, unfortunately. The idea behind set theory was to describe some universal building blocks of mathematics. The problem is that - unlike in the case of computable functions - such a universal building block can't exist.

There's a lot of mathematics which can't be formalized in classical first-order logic with the ZFC axioms. For most things you can get around this by adding enough Grothendieck universes, but this is still incomplete and always will be.