Mathematically speaking, what is an ‘Axiom’?
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– Math is like games. Axioms are the rules of a game.
– You can design a game with with whatever rules you want. In math, you can develop a theory with whatever axioms you want.
– However, not all rules make fun games. Likewise, not all axioms make interesting or useful mathematical theories.
People tend to use [ZFC set theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) as an example of an axiomatic theory, but I think it gives the misleading impression that axioms are always these low level rules that are “fundamental” and set in stone.
In practice, axioms are much more fluid than that and they exist in higher level theories: [groups](https://en.wikipedia.org/wiki/Group_(mathematics)), [rings](https://en.wikipedia.org/wiki/Ring_(mathematics)), and [vector spaces](https://en.wikipedia.org/wiki/Vector_space) have way more relevance in practical applications.
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