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Good Math by Mark C. Chu-Carroll

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Why?

There we are. That’s it: set theory in a nutshell. You can derive pretty much all of mathematics from those ten axioms plus simple first-order predicate logic. The integers fall out pretty naturally from the axiom of infinity; once you’ve got the integers, you can use the axiom of pairing to create the rationals; once you’ve got the rationals, you can use these axioms to derive Dedekind cuts to get the reals; once you’ve got the reals, you can use the axiom of replacement to get the transfinites. It just all flows out from these ten rules.

The amazing thing is that they’re not even particularly hard! The axioms make sense: the reason why each is needed is clear, and what each one means is clear. It doesn’t strain our brains to understand ...

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