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by thaumasiotes 1318 days ago
The answer isn't all that valid. ℵ₀ is just a name defined by the statement "the number of integers is ℵ₀". You could also call it Bob. If you don't think Bob would be a valid name, you should reject ℵ₀ too.

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.
2 comments
PebblesRox 1318 days ago
But I don't think Bob is a valid name because if I use it to refer to the cardinality of the naturals, nobody would know what I'm talking about.

I don't see why I have to reject aleph null just because I reject Bob.

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mbg721 1318 days ago
If it were called "Bob", students would inevitably read their profs' handwriting as "eight hundred six", leading to all sorts of hilarious confusion.
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elcomet 1318 days ago
> ℵ₀ is equal to ℶ₀

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.

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thaumasiotes 1318 days ago
> "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.

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.

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