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> The same way, I can just generate random real numbers, then define my set as in ascending order, and insert them in that order. To perform the diagonalizatio

> The same way, I can just generate random real numbers, then define my set as in ascending order, and insert them in that order. To perform the diagonalization, we can simply look at each real number that you randomly generate and add digits to the diagonalized number based on that order. It doesn't matter how much you shuffle them around afterward: the diagonalized number will still be different from all the generated numbers.

nico 4y ago
What you are saying is: 1) I will produce an infinite amount of numbers, then stop 2) I will look at the infinite amount of numbers and then produce a number which is not within the infinite numbers 3) Hey look, the number I produced in 2) is not within the infinite set, so there must be a larger infinite set The issue is that you can never finish 1). That’s the error/illusion. But if you get to cheat, then I can also add a step 4) As long as you add digits to my numbers to generate an additional number, I will reorder the set of generated real numbers and assign each of them a natural number. Of course the order will change from iteration to iteration. But in this way the generated real numbers will always have a corresponding natural number.
- LegionMammal978 4y ago
  Where did I imply 1)? We never finish producing the numbers, so unless our production order was fixed from the beginning, we never know the full value of the diagonalized number. But the partial diagonalized number (which we refine at each step by looking at the produced number) will gradually converge to a limit, which is a real number different from any number we will ever produce. At no point in the process do we know the limit exactly, but it is there nonetheless. > Of course the order will change from iteration to iteration. But in this way the generated real numbers will always have a corresponding natural number. The real numbers that you generate will always have a corresponding natural number. But the diagonalized number will never have a corresponding natural number, no matter how much you shuffle your generated values.
  - nico 4y ago
    You are saying that when you stop counting, the diagonal number will be different. That means that you stop. Which implies you are not going to infinity. If you are going to infinity, there is no side that “wins” (real or naturals). You can always keep generating an additional one for either side. The same way, we could continue this discussion forever if we just want to keep going back and forth (although HN does impose a limit of replies). And you could only make an assertion about the length of each side of the conversations, after the conversation ends. But if the conversation keeps going forever, you could never say to talked more than the other. Unless you specify a finite range for which you want to measure. Naturals and Reals are both infinite. Saying that the cardinality of one is different than the cardinality of the other, is just comparing the ideas of cardinality about them that we have come with, but their actual cardinality can never be known.
    - LegionMammal978 4y ago
      > You are saying that when you stop counting, the diagonal number will be different. Where, in all my comments, do I say that we must stop counting? You're putting words in my mouth here. Since we don't stop counting, we never know the diagonal number's full value (unless our order is predefined). All we know is that the diagonal number exists, and it is different from all the real numbers we will ever count. > If you are going to infinity, there is no side that “wins” (real or naturals). You can always keep generating an additional one for either side. Of course. I am not disputing this. If we count through two different sets in the same way, then the subsets of values we count through will have the same cardinality (equal to the cardinality of the set of natural numbers). But there's an important distinction. If we count through the natural numbers one-by-one without stopping, it's possible to exhaust the whole set of natural numbers, such that every single number will eventually get counted to. Meanwhile, if we count through the real numbers one-by-one, it is impossible to reach all of them. There will always be a bunch of real numbers that we will never ever count to. That's why we say that there are more real numbers than can be counted: there are uncountably many.
      - nico 4y ago

nico 4y ago

What you are saying is: 1) I will produce an infinite amount of numbers, then stop 2) I will look at the infinite amount of numbers and then produce a number which is not within the infinite numbers 3) Hey look, the number I produced in 2) is not within the infinite set, so there must be a larger infinite set The issue is that you can never finish 1). That’s the error/illusion. But if you get to cheat, then I can also add a step 4) As long as you add digits to my numbers to generate an additional number, I will reorder the set of generated real numbers and assign each of them a natural number. Of course the order will change from iteration to iteration. But in this way the generated real numbers will always have a corresponding natural number.

LegionMammal978 4y ago
Where did I imply 1)? We never finish producing the numbers, so unless our production order was fixed from the beginning, we never know the full value of the diagonalized number. But the partial diagonalized number (which we refine at each step by looking at the produced number) will gradually converge to a limit, which is a real number different from any number we will ever produce. At no point in the process do we know the limit exactly, but it is there nonetheless. > Of course the order will change from iteration to iteration. But in this way the generated real numbers will always have a corresponding natural number. The real numbers that you generate will always have a corresponding natural number. But the diagonalized number will never have a corresponding natural number, no matter how much you shuffle your generated values.
- nico 4y ago
  You are saying that when you stop counting, the diagonal number will be different. That means that you stop. Which implies you are not going to infinity. If you are going to infinity, there is no side that “wins” (real or naturals). You can always keep generating an additional one for either side. The same way, we could continue this discussion forever if we just want to keep going back and forth (although HN does impose a limit of replies). And you could only make an assertion about the length of each side of the conversations, after the conversation ends. But if the conversation keeps going forever, you could never say to talked more than the other. Unless you specify a finite range for which you want to measure. Naturals and Reals are both infinite. Saying that the cardinality of one is different than the cardinality of the other, is just comparing the ideas of cardinality about them that we have come with, but their actual cardinality can never be known.
  - LegionMammal978 4y ago
    > You are saying that when you stop counting, the diagonal number will be different. Where, in all my comments, do I say that we must stop counting? You're putting words in my mouth here. Since we don't stop counting, we never know the diagonal number's full value (unless our order is predefined). All we know is that the diagonal number exists, and it is different from all the real numbers we will ever count. > If you are going to infinity, there is no side that “wins” (real or naturals). You can always keep generating an additional one for either side. Of course. I am not disputing this. If we count through two different sets in the same way, then the subsets of values we count through will have the same cardinality (equal to the cardinality of the set of natural numbers). But there's an important distinction. If we count through the natural numbers one-by-one without stopping, it's possible to exhaust the whole set of natural numbers, such that every single number will eventually get counted to. Meanwhile, if we count through the real numbers one-by-one, it is impossible to reach all of them. There will always be a bunch of real numbers that we will never ever count to. That's why we say that there are more real numbers than can be counted: there are uncountably many.
    - nico 4y ago