User blog comment:BlauesWasser/Why Zero Shouldn't be considered a cardinal/@comment-30754445-20180501091716/@comment-30754445-20180506005334

Huh?

What I've just written is the standard definition of ordinals which can be found in any introductionary text on set theory.

And I don't get your objections. Surely you've heard of isomorphisms and homeomorphic embeddings? There are many many ways to map the concept of "natural numbers" into set theory. One of them, which is probably the simplest, is the one I've hinted to in my post:

0 = empty set

1 = {0}

2 = {0,1}

3 = {0,1,2}

4 = {0,1,2,3}

and so on.

There are other ways, of-course. What connects all of them, conceptually, is the fact that they all represent the same kind of structure. They all obey peano's axioms, for example. And in all of them, once we properly define addition and multiplicaiton, 2+2=4 and 12+34=56 and 12*34=408 and so on.

This is hardly a problem. You're free to choose any of these isomorphoic structures and call them "the set of natural numbers". Of-course you can't change your representation mid-way, but this just means that you need to be careful and rigorous when doing mathematics (which should go without saying, anyway).

By the way, in analysis and algebra, when people use the symbols ℕ and ℚ, they are referring to subsets of ℝ (and they don't particularly care which model of ℝ they're using). This is the only interpertation by which those fields of mathematics can make any sense. So yes, under any reasonable interpertation of these symbols, the statement ℕ⊆ℚ is technically correct.