Is there some axiom or postulate that defines addition?
I've always wondered this.
Well, when you get to that level it becomes rather difficult. I believe that if you define the natural numbers as an ordered set, you can define an operation on them that essentially says "go to the 'next' number in the set", which we can call "succession" and denote as s(n), where n is a natural number. In normal terms, s(n)=n+1, but since we haven't defined "addition" yet, we can't really call it that yet. Then, you can define addition as repeated succession in some way, and from there define 1+1 and find it to be equal to 2.
At least, I think the above method can work. Obviously someone better than I would have to come along and codify it.
You do not really need to delve into set theory, unless you want to prove that the rules we set up for integer addition follow from the rules we have from set axioms (i.e, mathematical reductionism).
You can, of course, DEFINE 2 to be equal to 1+1.
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