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It's not obvious how to order the rationals either, but there are of course loads of orderings that work, and it turns out that they can easily be put into corr

It's not obvious how to order the rationals either, but there are of course loads of orderings that work, and it turns out that they can easily be put into correspondence with the naturals. Such orderings are not from lowest to highest, obviously, but they exist, and you might naturally assume there's one for the reals. The thing with the reals is that no orderings work, that's what Cantor's diagonalization proof does.

nico 4y ago
As long as you have time and you keep counting, you can count anything. But eventually you will have to stop. Counting infinity is a contradiction, as you can never stop. Hence you can never get a final output. It doesn’t make sense that the cardinality of the set of natural numbers is some finite symbol (aleph-null). We can never truly know the cardinality of that set, because we can’t count to infinity. There’s an additional issue here. Which is: what does it mean to count? Because counting natural numbers usually means generating them in a certain order. The issue is that we cant come up with a way to produce reals in an orderly way one after the other. But we can always generate a random real number, then add it to a set in a position between a smaller and a bigger number. That way the set will always be in order, even if the number wasn’t produced right after a smaller number and before a bigger number. The order is essentially ad-hoc and varies as you generate the set. And if you kept going “for infinity” you would generate all the real numbers. But we can never count to infinity, not even for the natural numbers.
- denton-scratch 4y ago
  "Counting", for these purposes, doesn't mean determining how many there are; it means going through them, or a subset of them, in some defined order.
  - nico 4y ago
    You can go through all the reals one by one, in order. As long as you can keep going for infinity. Simple: at each step generate a random real number and assign it a natural number. You will never run out of natural numbers. And if you keep going to infinity, then the probability of generating all reals is 1. So you would be generating all reals and assigned all of them a natural number. The issue is that the proofs for uncountability assume we can finish counting natural numbers and after finishing, generate a real that is not in the naturals. But if we had finished counting, we could do the same for the naturals, just add 1 to the largest number and you’d have something not in the set. Of course you can only do that if you have a finite set, which is not the case for the naturals =><=
    - denton-scratch 4y ago
      > You can go through all the reals one by one, in order. As long as you can keep going for infinity. > Simple: at each step generate a random real number and assign it a natural number. This is nonsense. You said you were going to go through the reals "in order". Then you give a procedure that involves generating reals randomly. Just think about what you said. It's not possible to go through the reals "in order"; that would imply that for any real, there is a "next" real. Let the starting real be A. Suppose there is a next real; let it be B (generate it using some random process, if that floats your boat). The interval between them, B-A, let us call that C. I can divide C by 2, and produce a new real D=A+(C/2), which is smaller than B and larger than A. Therefore B is not the next real. This is true for any choice of A and B, so it follows that there cannot be a "next" real, and so it's impossible to go through the reals one by one, in order. > As long as you can keep going for infinity. That's irrelevant, even if you restrict yourself to the reals between 0.001 and 0.002, because whatever process you use to select the "next" real, I can find one that is a better candidate, by dividing the difference by two. The number of reals between 0.001 and 0.002 is the same as the total number of reals, and it's bigger than the number of natural numbers. That's a mind-boggling conclusion; but note that Cantor died with his mind well and truly boggled (he went mad).
      - nico 4y ago
        > Let the starting real be A. Suppose there is a next real; let it be B (generate it using some random process, if that floats your boat). The interval between them, B-A, let us call that C. I can divide C by 2, and produce a new real D=A+(C/2), which is smaller than B and larger than A. Therefore B is not the next real. You need to decide whether you want to define counting as “there being a next element”, or “being able to produce those elements in a finite way with a predefined order”. The issue here is that you are trying to make a finite/determinate assertion about an infinite process. There are no reals in between 0.001 and 0.002, unless you generate them. And when you do, you can count them one by one until you finish generating them. Just like you could never finish counting infinite real numbers, you could never finish counting infinite natural numbers. So then what Cantors proof is saying is that if you could somehow count to infinity and get to a result, some infinities would have more elements than others. But that is a contradiction, because you can’t finish counting to infinity. Hence, you cannot know infinity nor its cardinality. Not for the real numbers and not for the natural numbers either.

nico 4y ago

As long as you have time and you keep counting, you can count anything. But eventually you will have to stop. Counting infinity is a contradiction, as you can never stop. Hence you can never get a final output. It doesn’t make sense that the cardinality of the set of natural numbers is some finite symbol (aleph-null). We can never truly know the cardinality of that set, because we can’t count to infinity. There’s an additional issue here. Which is: what does it mean to count? Because counting natural numbers usually means generating them in a certain order. The issue is that we cant come up with a way to produce reals in an orderly way one after the other. But we can always generate a random real number, then add it to a set in a position between a smaller and a bigger number. That way the set will always be in order, even if the number wasn’t produced right after a smaller number and before a bigger number. The order is essentially ad-hoc and varies as you generate the set. And if you kept going “for infinity” you would generate all the real numbers. But we can never count to infinity, not even for the natural numbers.

denton-scratch 4y ago
"Counting", for these purposes, doesn't mean determining how many there are; it means going through them, or a subset of them, in some defined order.
- nico 4y ago
  You can go through all the reals one by one, in order. As long as you can keep going for infinity. Simple: at each step generate a random real number and assign it a natural number. You will never run out of natural numbers. And if you keep going to infinity, then the probability of generating all reals is 1. So you would be generating all reals and assigned all of them a natural number. The issue is that the proofs for uncountability assume we can finish counting natural numbers and after finishing, generate a real that is not in the naturals. But if we had finished counting, we could do the same for the naturals, just add 1 to the largest number and you’d have something not in the set. Of course you can only do that if you have a finite set, which is not the case for the naturals =><=
  - denton-scratch 4y ago
    > You can go through all the reals one by one, in order. As long as you can keep going for infinity. > Simple: at each step generate a random real number and assign it a natural number. This is nonsense. You said you were going to go through the reals "in order". Then you give a procedure that involves generating reals randomly. Just think about what you said. It's not possible to go through the reals "in order"; that would imply that for any real, there is a "next" real. Let the starting real be A. Suppose there is a next real; let it be B (generate it using some random process, if that floats your boat). The interval between them, B-A, let us call that C. I can divide C by 2, and produce a new real D=A+(C/2), which is smaller than B and larger than A. Therefore B is not the next real. This is true for any choice of A and B, so it follows that there cannot be a "next" real, and so it's impossible to go through the reals one by one, in order. > As long as you can keep going for infinity. That's irrelevant, even if you restrict yourself to the reals between 0.001 and 0.002, because whatever process you use to select the "next" real, I can find one that is a better candidate, by dividing the difference by two. The number of reals between 0.001 and 0.002 is the same as the total number of reals, and it's bigger than the number of natural numbers. That's a mind-boggling conclusion; but note that Cantor died with his mind well and truly boggled (he went mad).
    - nico 4y ago
      > Let the starting real be A. Suppose there is a next real; let it be B (generate it using some random process, if that floats your boat). The interval between them, B-A, let us call that C. I can divide C by 2, and produce a new real D=A+(C/2), which is smaller than B and larger than A. Therefore B is not the next real. You need to decide whether you want to define counting as “there being a next element”, or “being able to produce those elements in a finite way with a predefined order”. The issue here is that you are trying to make a finite/determinate assertion about an infinite process. There are no reals in between 0.001 and 0.002, unless you generate them. And when you do, you can count them one by one until you finish generating them. Just like you could never finish counting infinite real numbers, you could never finish counting infinite natural numbers. So then what Cantors proof is saying is that if you could somehow count to infinity and get to a result, some infinities would have more elements than others. But that is a contradiction, because you can’t finish counting to infinity. Hence, you cannot know infinity nor its cardinality. Not for the real numbers and not for the natural numbers either.