3 ms·

There are the same number of even integers as there are even and odd integers

by petesergeant 4y ago

charcircuit 4y ago
If they are the same number what number is it.
- tsimionescu 4y ago
  It's called ℵ 0 (Aleph 0).
  - elcomet 4y ago
    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.

There are the same number of even integers as there are even and odd integers

charcircuit 4y ago

If they are the same number what number is it.

tsimionescu 4y ago
It's called ℵ 0 (Aleph 0).
- elcomet 4y ago
  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.