5 ms·

Given that a one-to-one correspondence is just a renaming, and renaming things doesn't change how many there are, this seems sound.

cubefox 4y ago
It's not just a renaming. The following is true: The square numbers are a proper subset of the natural numbers. If you switch "square numbers" and "natural numbers" it is false. If two terms can't be exchanged salva veritate, they are not equivalent.
- firstlink 4y ago
  Your failure to do the actual renaming doesn't make the renaming a failure. No one has claimed that the set of squares is extensionally equal to the naturals. The existence of a bijection has nothing to do with extensional equality of the given sets. It's not even a particularly strong equivalence among those in everyday use. I guess the category theorists have a cute name for this (morphism? Who knows...), but it is to set cardinality as homomorphism are to groups. But in fact even if the claim had been about extensional equality, HoTT shows how this can be carried out correctly: the equivalence between two types (which is supposed to be actual Leibniz equality, not merely equinumerosity as in the example) must be used, as if a function, to transport actual terms between the two types. Using this mechanism, the subtype of square naturals is in fact extensionally equal to the naturals. For example, the term `f 4` where `f` is the transport along `squares = N` is in fact the sum of the multiplicative identity with itself. This is what I meant by actually doing the renaming.
  - cubefox 4y ago
    He said "just" a renaming, which suggests just the names change while the meaning stays the same, which isn't the case. Maybe he meant it in some different "HoTT" sense, but he clearly said > Given that a one-to-one correspondence is just a renaming, and renaming things doesn't change how many there are, this seems sound. which would only be the case for renaming salva veritate -- or by begging the question in favor of one-to-one correspondence being sufficient for something being the "same size" and against proper subsethood being sufficient for something being not the same size. But that's exactly the question when arguing for or against Hume's Principle, not something which can be assumed.
    - thaumasiotes 4y ago
      > He said "just" a renaming, which suggests just the names change while the meaning stays the same, which isn't the case. Huh? That's exactly what's happening. If you rename 1, 2, and 3 to one, two, and three, the names have changed and the meanings haven't. As you note, this is what "renaming" means. It used to be true that 1 + 2 = 3, but now that's gibberish and one + two = three instead. Nothing about that changes if you instead rename 1, 2, and 3 to 88, 89, and 90. It will still be true that 88 + 88 = 89. You don't get to reinterpret what the name means after you assign it.
      - cubefox 4y ago
        Well, then the squares are still a proper subset of the natural numbers, while still being in a one-to-one correspondence, so Galileo's Paradox remains. Then renaming didn't change anything.
- gmane 4y ago
  The problem is that infinity is not a number. Infinity plus one is equal to infinity. This is not a trait that any number has. There is no number where you can add one and get the same number. The subset of an infinite set can itself be infinite. If you can map bi-directionality (which you can for square numbers and naturals) between an infinite set and its' infinite subset, then they are equivalent in size. Your problem is that you expect infinity (well, one type of infinity, there are ordered infinities [edit: when I said infinities, I meant infinite numbers, which the common infinite is not an infinite number, it's infinity], but that's outside our scope here) to be ordered, which it is not. Numbers are ordered, but infinity is not a number and is not ordered in the numbers.
  - thaumasiotes 4y ago
    > The problem is that infinity is not a number. Infinity plus one is equal to infinity. This is not a trait that any number has. There is no number where you can add one and get the same number. This is all false. What you describe is a violation of the Archimedean principle. "Numbers" are a wider concept than the Archimedean principle.
- tshaddox 4y ago
  It’s still just a renaming. In fact you might even perform the renaming and thus form the bijection without even realizing it, if you talked about “the first square number,” “the second square number,” etc.
  - quickthrower2 4y ago
    It is an injection, but not a surjection. You need neither though. For example: y = x ^ 2 when even, (x+1) ^ 2 otherwise
    - thaumasiotes 4y ago
      > It is an injection, but not a surjection. It's a bijection. (And therefore obviously also a surjection.) "One-to-one correspondence" is a correct, but incomplete, description.
      - quickthrower2 4y ago
        ∄ x ∈ ℕ: x² = 5 Unless you consider the domain of f the "renamed" range, in which case yes tautologically it is bijective.
        mb7733 4y ago
        You're talking past one another. The other poster is discussing a correspondence between N and S, where S is the set of squares. So the function being discussed is f: N -> S, f(n) = n^2, which is obviously bijective. You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. Any surjective function can be made non-surjective by considering a larger range.
        thaumasiotes 4y ago
        > You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. It's worse than just irrelevant. The claim is that we have two sets, ℕ and S, that are the same size. The argument that they are the same size is the observation that we've demonstrated a bijection between them. No one ever mentioned a function from ℕ to ℕ, because that would tell you nothing about S. It would be like saying "in order to show that 5 is odd, observe that 2 and 2 are both even". The idea is gibberish.
- quickthrower2 4y ago
  We are only taking sizes here not “is it a group” etc. So let’s play that old 21s drinking game, but infinitely instead of 1 to 21, and rename 1 to 1, 2 to 4, 3 to 9 and so on, up to x for any x. Same set, different lingo. That you can’t map back 5 is irrelevant although kind of leaves a bad taste.

cubefox 4y ago

It's not just a renaming. The following is true: The square numbers are a proper subset of the natural numbers. If you switch "square numbers" and "natural numbers" it is false. If two terms can't be exchanged salva veritate, they are not equivalent.

firstlink 4y ago
Your failure to do the actual renaming doesn't make the renaming a failure. No one has claimed that the set of squares is extensionally equal to the naturals. The existence of a bijection has nothing to do with extensional equality of the given sets. It's not even a particularly strong equivalence among those in everyday use. I guess the category theorists have a cute name for this (morphism? Who knows...), but it is to set cardinality as homomorphism are to groups. But in fact even if the claim had been about extensional equality, HoTT shows how this can be carried out correctly: the equivalence between two types (which is supposed to be actual Leibniz equality, not merely equinumerosity as in the example) must be used, as if a function, to transport actual terms between the two types. Using this mechanism, the subtype of square naturals is in fact extensionally equal to the naturals. For example, the term `f 4` where `f` is the transport along `squares = N` is in fact the sum of the multiplicative identity with itself. This is what I meant by actually doing the renaming.
- cubefox 4y ago
  He said "just" a renaming, which suggests just the names change while the meaning stays the same, which isn't the case. Maybe he meant it in some different "HoTT" sense, but he clearly said > Given that a one-to-one correspondence is just a renaming, and renaming things doesn't change how many there are, this seems sound. which would only be the case for renaming salva veritate -- or by begging the question in favor of one-to-one correspondence being sufficient for something being the "same size" and against proper subsethood being sufficient for something being not the same size. But that's exactly the question when arguing for or against Hume's Principle, not something which can be assumed.
  - thaumasiotes 4y ago
    > He said "just" a renaming, which suggests just the names change while the meaning stays the same, which isn't the case. Huh? That's exactly what's happening. If you rename 1, 2, and 3 to one, two, and three, the names have changed and the meanings haven't. As you note, this is what "renaming" means. It used to be true that 1 + 2 = 3, but now that's gibberish and one + two = three instead. Nothing about that changes if you instead rename 1, 2, and 3 to 88, 89, and 90. It will still be true that 88 + 88 = 89. You don't get to reinterpret what the name means after you assign it.
    - cubefox 4y ago
      Well, then the squares are still a proper subset of the natural numbers, while still being in a one-to-one correspondence, so Galileo's Paradox remains. Then renaming didn't change anything.
gmane 4y ago
The problem is that infinity is not a number. Infinity plus one is equal to infinity. This is not a trait that any number has. There is no number where you can add one and get the same number. The subset of an infinite set can itself be infinite. If you can map bi-directionality (which you can for square numbers and naturals) between an infinite set and its' infinite subset, then they are equivalent in size. Your problem is that you expect infinity (well, one type of infinity, there are ordered infinities [edit: when I said infinities, I meant infinite numbers, which the common infinite is not an infinite number, it's infinity], but that's outside our scope here) to be ordered, which it is not. Numbers are ordered, but infinity is not a number and is not ordered in the numbers.
- thaumasiotes 4y ago
  > The problem is that infinity is not a number. Infinity plus one is equal to infinity. This is not a trait that any number has. There is no number where you can add one and get the same number. This is all false. What you describe is a violation of the Archimedean principle. "Numbers" are a wider concept than the Archimedean principle.
tshaddox 4y ago
It’s still just a renaming. In fact you might even perform the renaming and thus form the bijection without even realizing it, if you talked about “the first square number,” “the second square number,” etc.
- quickthrower2 4y ago
  It is an injection, but not a surjection. You need neither though. For example: y = x ^ 2 when even, (x+1) ^ 2 otherwise
  - thaumasiotes 4y ago
    > It is an injection, but not a surjection. It's a bijection. (And therefore obviously also a surjection.) "One-to-one correspondence" is a correct, but incomplete, description.
    - quickthrower2 4y ago
      ∄ x ∈ ℕ: x² = 5 Unless you consider the domain of f the "renamed" range, in which case yes tautologically it is bijective.
      - mb7733 4y ago
        You're talking past one another. The other poster is discussing a correspondence between N and S, where S is the set of squares. So the function being discussed is f: N -> S, f(n) = n^2, which is obviously bijective. You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. Any surjective function can be made non-surjective by considering a larger range.
        thaumasiotes 4y ago
        > You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. It's worse than just irrelevant. The claim is that we have two sets, ℕ and S, that are the same size. The argument that they are the same size is the observation that we've demonstrated a bijection between them. No one ever mentioned a function from ℕ to ℕ, because that would tell you nothing about S. It would be like saying "in order to show that 5 is odd, observe that 2 and 2 are both even". The idea is gibberish.
quickthrower2 4y ago
We are only taking sizes here not “is it a group” etc. So let’s play that old 21s drinking game, but infinitely instead of 1 to 21, and rename 1 to 1, 2 to 4, 3 to 9 and so on, up to x for any x. Same set, different lingo. That you can’t map back 5 is irrelevant although kind of leaves a bad taste.