Sure. You can do that with the irrationals as well. Whenever you have a new irrational, I'll increment my counter and assign it thusly. You show that there's i

Sure. You can do that with the irrationals as well. Whenever you have a new irrational, I'll increment my counter and assign it thusly. You show that there's irrationals between two assigned ones that were missed and I will continue to increment the counter accordingly and assign it to the missed irrationals. Then we will have our mapping to the whole numbers like you claimed. The point is that rule is insufficient to demonstrate the distinction we wish for. There's other ones such as the diagonalization or the column proof, you cannot do these directly with the primes, only with this middle step of assigning them a cardinality If you can only do them if you abstract your representation and assume you can assign them like that which I've showed, then you can do for any other set and it makes it no longer valid. It's only about what proofs can be demonstrated and what proofs cannot directly. Cantor's work is, in this way, more about the nature of the "generators" of the sequences then the sequences themselves and categorizing them like algorithmic complexity. The act of mapping is contingent on the sequences peano successor being algebraic. When you assign them whole number cardinality, you apply another "generator" that normalizes the thing you're inspecting and removes it from inspection. There's a distinction here that's not appreciated. But I've honestly lost hope