I guess one benefit is that one could prove the existance of all the numbers inductively ... but that doesn't seem especially exciting.

That's probably because they made the mistake of teaching you arithmetic first, so you take it for granted. I think the idea behind "new math", back in the 1960s, was that Zermelo-Fraenkel set theory would be taught before arithmetic, since it was more fundamental. Then arithmetic would seem like something new and wonderful when the little four year olds derived it from set theory.

Unfortunately for me, new math didn't hit until I was in grade school, so I had the same reaction that you did. Thirty years later, I've rejected the axiom of choice as well.

Nick George
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but, what I'm wondering is, can one prove that 2 + 3 = 5 using this theory?

I read the article on the Axiom of Choice a few times, but I'm still a little fuzzy on it, I'm working on it.