User blog comment:BlauesWasser/Why Zero Shouldn't be considered a cardinal/@comment-30754445-20180501091716/@comment-30754445-20180506180526

Now I've completely lost you.

If you want to view ordinals as "isomorphism classes of well-orderings" then ordinals aren't even sets. How is that any better than equating ordinals with certain sets?

Since finite ordinals are isomorphic to ℕ, you haven't found a way to bypass the concept of ℕ′ .You still have two distinct models of the naturals (the finite ordinals and whatever set you decided to call ℕ), which is precisely the thing you wanted to avoid.

And I still don't understand all your talk about "structure". What the **** are you talking about? The topology of ℕ (which is discrete) doesn't change with your specific model. The algebric properties of ℕ don't change either. And if something is difficult to prove with one model, there's nothing to stop you from using the concept of isomorphism to "translate" the problem into an easier one in another model.

As for your question:

In my view, the reals (and rationals, and naturals) are represented by a unique concept. You can define this concept, for example, by writing down a list of axioms that it obeys. And since there's no denying that there are many sets which are accurate models of these axioms, it makes no sense to insist that the reals/rationals/naturals are tied to a unique set.

Not in any fundamental way, at any rate.

We can, of-course, decide - for sake of convinience - to pick one of these models and call it "the set of the reals". It doesn't really matter, though, because the model itself is unimportant. The concept of "fiveness", say, isn't tied to a particular set. A fact that you've admitted yourself, the moment you've defined the ordinal 5 as "the isomorphism class of well-orders in which {0,1,2,3,4} is a member"