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>What separates math from pure logic is that math has a set of very intuitive axioms.

Taking certain sets of axioms can lead to very un-intuitive conclusions, that's why some people care about building a solid foundations. A classic example is https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox , which states "Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball." If we change the foundations to remove the axiom of choice, this paradox (and many others like it) is destroyed.

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