3 ms·

"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.
    - denton-scratch 4y ago
      > There are no reals in between 0.001 and 0.002, unless you generate them. You seem to be thinking of a real as a string of digits, or a block of bits, which is just the representation of a real. Like, a number doesn't exist until you think of it. That view doesn't help much in thinking about these matters. > 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. So you're saying Cantor's "proof" embodies a contradiction. You must be a very great mathematician. I'm not a mathematician, so I'd better just bow out now. Anyway, I'm way too old to start thinking of numbers as some kind of generative process.
      - nico 4y ago
        Why would someone need to be a mathematician to think and understand? That’s just gate keeping through labels. And probably the reason why so many old ideas go completely unchecked and are taken as gospel for so long with no one questioning them. You can clearly reason about this stuff, even if you are not a mathematician. But maybe it scares you to think that something that you thought was “true”, is just someone else’s opinion about what they thought it was true.

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.
  - denton-scratch 4y ago
    > There are no reals in between 0.001 and 0.002, unless you generate them. You seem to be thinking of a real as a string of digits, or a block of bits, which is just the representation of a real. Like, a number doesn't exist until you think of it. That view doesn't help much in thinking about these matters. > 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. So you're saying Cantor's "proof" embodies a contradiction. You must be a very great mathematician. I'm not a mathematician, so I'd better just bow out now. Anyway, I'm way too old to start thinking of numbers as some kind of generative process.
    - nico 4y ago
      Why would someone need to be a mathematician to think and understand? That’s just gate keeping through labels. And probably the reason why so many old ideas go completely unchecked and are taken as gospel for so long with no one questioning them. You can clearly reason about this stuff, even if you are not a mathematician. But maybe it scares you to think that something that you thought was “true”, is just someone else’s opinion about what they thought it was true.