> In other words, if you want to work backwards and say that smaller sets (like the natural numbers) are a subset of the bigger sets (like complex numbers), then you have to pick a “biggest set” containing all numbers, which is unsatisfactory. Somebody always wants a bigger set.
Exactly what we did in the Analysis I course I attended during my bachelor: defined the reals axiomatically, and the N as smallest inductive(?) subset containing 0.
Satisfactory or not, it worked well for the purpose. And I actually liked this definition, if anything because it was original. Mathematical definitions don't need to have some absolute philosophical value, as long as you prove that yours is equivalent to everyone else's it's fine.
> as long as you prove that yours is equivalent to everyone else's it's fine
That’s exactly the point I was making in the first place.
“Unsatisfactory” just means “unsatisfactory” in the sense that some mathematicians out there won’t be able to use your definitions and still get the subset property. This means that you are, in all realities, forced to deal with the separate notions of “equivalence” and “equality”. Which is what the article is talking about—all I’m really saying here is that you can’t sidestep equivalence by being clever.
Exactly what we did in the Analysis I course I attended during my bachelor: defined the reals axiomatically, and the N as smallest inductive(?) subset containing 0.
Satisfactory or not, it worked well for the purpose. And I actually liked this definition, if anything because it was original. Mathematical definitions don't need to have some absolute philosophical value, as long as you prove that yours is equivalent to everyone else's it's fine.