10 ms·

I like when someone asks a question that looks snarky, maybe rhetorical like saying "what you are saying is nonsense, look, this question cannot be answered", b

I like when someone asks a question that looks snarky, maybe rhetorical like saying "what you are saying is nonsense, look, this question cannot be answered", but then there is a perfectly valid answer

thaumasiotes 4y 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.
- PebblesRox 4y 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.
  - mbg721 4y ago
    If it were called "Bob", students would inevitably read their profs' handwriting as "eight hundred six", leading to all sorts of hilarious confusion.
- elcomet 4y 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.
  - thaumasiotes 4y 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.

thaumasiotes 4y 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.

PebblesRox 4y 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.
- mbg721 4y ago
  If it were called "Bob", students would inevitably read their profs' handwriting as "eight hundred six", leading to all sorts of hilarious confusion.
elcomet 4y 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.
- thaumasiotes 4y 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.