[RHSeeger]

Is addition defined _by_ set theory, or is set theory one way of defining addition? If it's the later, then there could be other ways of defining addition that don't have the same results for infinity (because our math system doesn't really "work" for infinity, or 0, depending on the circumstances).

I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that; a question.

[sritchie]

It's the latter; I'm also not a mathematician, just a guy who worked through Halmos's "Naive Set Theory" in intense detail...

But your question actually hints at my most profound takeaway from that whole book. I think what you're saying is right, AND that foundations-of-mathematics folks spent a long intense period searching for different set theory axioms that did NOT lead to transfinite numbers. But anything anyone could come up with that included "the axiom of infinity" led to transfinites leaking in.

Which begs the question of how to think about these things. Are they "real"? Are they an oddball side effect that we shouldn't take seriously?

I think you've arrowed right to the philosophical heart of all of this.

[chevman]

Does everything become a paradox given enough time and/or thought?

I think we often end up at the end of logical thought processes back at the original question - how can we observe and describe a system that we are inherently a part of?

[gowld]

There are many ways of definiting everything. Most of them are equivalent in the ways that matter, which is why math "works" so well as the language of science. Some of them are different in critical ways, which opens up vistas of new objects and concepts.