The nature of number

[jonb]

So uh--in terms of the "different types of sets" thing... Yes, the relations we put on sets is the stuff that's actually interesting, but the semantics of modern math insists that those relations come after the existence of the sets (and hence, within that framework, these structures can't say very much about the nature of sets themselves). The rationals look very different from the perspectives of regular and italicized additions, but it's the same set (the same collection of symbols, objects, whatever) at the end of the day, right?

Afterthought: damn, that was a long rant.

Long but good

but the semantics of modern math insists that those relations come [i]after the existence of the sets [/i]

Ah This is where I am having my problem. It seems to me one form of set has those relations inherently built into it where as the other form those relations are not originally expressed.