The name isn't unique either; by definition, ℵ₀ is equal to ℶ₀. This should be a clue that the term ℵ₀ is not actually meant to identify the number in question. Rather, what's going on is that there is a conceptual system of ℵ numbers, and another conceptual system of ℶ numbers, and the number at index 0 in each of those systems is the cardinality of the naturals.
I don't see why I have to reject aleph null just because I reject Bob.
That's the definition of ℶ₀. So of course they are equal.
"Two" is also just a name for the successor of 1. I could also call it "bob" and thus "two" would not be unique, but I don't see the point. The fact is that the cardinal of countable numbers is a mathematical concept which has a name, and can be manipulated. Which is what matters, and what the parent poster maybe did not understand.
There is no point, because 2 is well defined.
That is also true of the ℶ numbers, but it is not true of the ℵ numbers. That's why, in that case, it's necessary to have multiple names for the same number.
> The fact is that the cardinal of countable numbers is a mathematical concept which has a name, and can be manipulated.
No, that's not a fact, that's what I'm saying.
The cardinality of the naturals is a mathematical concept. It is referred to as "the cardinality of the naturals", or by many similar phrases.
But it is not referred to by the name ℵ₀. ℵ₀ is a name that refers to a different concept, the cardinality of the ordinal number ω. The two cardinalities are equal, but ℵ₀ specifically refers to one of them, conceptually, rather than the other.
ℶ₀ refers to a different concept again. That's the one that is meant to be manipulable.
