User blog comment:BlauesWasser/Why Zero Shouldn't be considered a cardinal/@comment-30754445-20180501091716/@comment-27513631-20180506111804

I'm not saying that what you've said is wrong, I'm saying it's unnatural.

Ordinals aren't transitive sets of transitive sets; that's just how they're defined in set theories with excluded middle and epsilon induction, like ZFC.

Ordinals are isomorphism classes of wellorderings.

You can define \(\nathbb N\) as a subset of \(\mathbb R\), and that's fine, especially if you're fine with disregarding the usage of \(\mathbb N'\) to define the reals. This does obfuscate the stand-alone structure of the naturals, as it realises them as a subcollection of the reals, but that's fine. Kinda. You may still have the wrong amount of structure, as there are multiple order-embeddings, multiple monoid-embeddings, you still have the topology of the reals, and you have to prove closure under semiring operations as opposed to getting them automatically. It's not the only way to make them make sense, though. Just view the 'subsets' as monos (injections) in the appropriate categories, and make these coercions implicit. I believe this better respects that they all have distinct structures.

And. You can change your representation midway. It's not a problem at all, really. You just have to ensure that all of your properties are isomorphism-invariant, which is externally true in theories like ETCS, internally true in HoTT, and has to be considered on a case-by-case basis in material set theories like ZFC and KP.