5 ms·

We're talking about infinity, not practical considerations. > (or insert them in order), There is no order. You can't describe one, and it is proven that no o

by hgsgm 4y ago

We're talking about infinity, not practical considerations. > (or insert them in order), There is no order. You can't describe one, and it is proven that no order exists. > you can count as many real numbers as you want No, you can't count more than 0% of them, even in infinite steps.

nico 4y ago
The order is arbitrary. What is the order of natural numbers? If you say they are in ascending order, that’s just the order in which you count them. The same way, I can just generate random real numbers, then define my set as in ascending order, and insert them in that order. Which is what we essentially do with natural numbers as well. Except that the order is a sort of mainstream standard and has been drilled into our minds since babies.
- LegionMammal978 4y ago
  > 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.
- haneul 4y ago
  Ascending order is just one of many arbitrary orders. An ordering is a first output of a generator function, a second output, a third output, etc, where for any arbitrarily chosen element X of the set (such as the positive integers or reals), we can show that the generator will reach X in N steps, for some finite N. That is why ascending order is the norm for the positive integers. For any positive integer X, it is clear that we will reach it in X steps using that generator function. So, it’s not a matter of being mainstream, it’s just the simplest. The difference between the infinity of the reals and the positive integers is that there is no such generator function for the reals. No matter what function you give me, I can write a finite number that you will not reach in a finite number of steps. Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps. On the other hand, try giving me a finite positive integer that I cannot reach in finite steps.
  - nico 4y ago
    Neither the reals nor the naturals “exist”. We produce them as we need them. Reals and naturals both have the same number of symbols: unknowable (“infinity”). > Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps. f(n)=n If you give n to that formula, it will generate n in 1 step. > On the other hand, try giving me a finite positive integer that I cannot reach in finite steps. Can’t come up with one. As long as is finite you can always just output the number itself. Anything finite is countable. Anything infinite is unknowable.
    - haneul 4y ago
      Nono. F(n) where n is the number of steps, not the desired number. You’ve got it backwards. The number is the output, not the input.
      - nico 4y ago
        N was the number of steps, but sure: F(x)=x, will give you x in one step.

by hgsgm 4y ago

nico 4y ago

The order is arbitrary. What is the order of natural numbers? If you say they are in ascending order, that’s just the order in which you count them. The same way, I can just generate random real numbers, then define my set as in ascending order, and insert them in that order. Which is what we essentially do with natural numbers as well. Except that the order is a sort of mainstream standard and has been drilled into our minds since babies.

LegionMammal978 4y ago
> 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.
haneul 4y ago
Ascending order is just one of many arbitrary orders. An ordering is a first output of a generator function, a second output, a third output, etc, where for any arbitrarily chosen element X of the set (such as the positive integers or reals), we can show that the generator will reach X in N steps, for some finite N. That is why ascending order is the norm for the positive integers. For any positive integer X, it is clear that we will reach it in X steps using that generator function. So, it’s not a matter of being mainstream, it’s just the simplest. The difference between the infinity of the reals and the positive integers is that there is no such generator function for the reals. No matter what function you give me, I can write a finite number that you will not reach in a finite number of steps. Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps. On the other hand, try giving me a finite positive integer that I cannot reach in finite steps.
- nico 4y ago
  Neither the reals nor the naturals “exist”. We produce them as we need them. Reals and naturals both have the same number of symbols: unknowable (“infinity”). > Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps. f(n)=n If you give n to that formula, it will generate n in 1 step. > On the other hand, try giving me a finite positive integer that I cannot reach in finite steps. Can’t come up with one. As long as is finite you can always just output the number itself. Anything finite is countable. Anything infinite is unknowable.
  - haneul 4y ago
    Nono. F(n) where n is the number of steps, not the desired number. You’ve got it backwards. The number is the output, not the input.
    - nico 4y ago
      N was the number of steps, but sure: F(x)=x, will give you x in one step.