4 ms·

If as you say, the infinity of the real numbers lacks the property of predefinable order, whereas the infinity of the natural numbers does have that property, d

by haneul 4y ago

If as you say, the infinity of the real numbers lacks the property of predefinable order, whereas the infinity of the natural numbers does have that property, does it not strike you that those infinities may be fundamentally different in some way relating to size, and therefore mapability?

nico 4y ago
It only strikes me that we are trying so hard to know the unknowable. Infinite essentially mean unknown. How could you possible compare the size of two unknowns? You just can’t, but we want to be able to reduce something infinite/unknowable, to a finite symbol that we can use in our limited language. Edit: for sibling comment -> how would you know the cardinality of a set without counting it? And if you have two infinite sets, how would you ever finish counting them to determine their cardinality and compare them? It doesn’t even make sense to talk about the cardinality of an infinite set, infinity is not a number. We pretend that the infinity symbol can somehow magically summarize and quantize something that is essentially unknowable. The infinity symbol doesn’t make something magically finite. The cardinality of an infinite set is unknowable. It is not “infinity”.
- LegionMammal978 4y ago
  > Infinite essentially mean unknown. How could you possible compare the size of two unknowns? The point is, that's not really what infinite means. A set being infinite just means that we can start listing elements, and we'll always be able to find a new element that we haven't listed before. To compare two infinite sets A and B, we can consider functions F that take any element from A and output some element from B. If A and B have the same cardinality, then we can always describe a certain function F that can potentially output any element from B, if we know the right input from A. But if B has a greater cardinality than A, then we cannot find any such function. No matter which function F we choose, there will always be certain values in B that can never be output by F given any input in A. Thus, we say that B has "more elements" than A in a certain sense. In the case of Cantor's argument, A is the set of natural numbers, B is the set of real numbers, and F is the ordering we choose. A variation of Cantor's argument can be expressed fully in terms of finite objects. Suppose we have a function F such that F(i) = f_i, where f_i is a function such that f_i(j) represents the jth digit of the ith real number in the sequence. Then, we can write a new function g(j) = (F(j)(j) + 2) mod 10, or some variation. This function g(j) similarly represents a real number. Now, we can take any i we want and start comparing f_i(1) vs. g(1), f_i(2) vs. g(2), etc. After a finite amount of time, we'll always reach a point where the two functions differ. This means that the two functions represent two different real numbers. Therefore, the real number represented by g is not represented by any of the f_i, no matter which function F we start with.
  - nico 4y ago
    Thank you, I really appreciate your thoughtful explanation. > A set being infinite just means that we can start listing elements, and we'll always be able to find a new element that we haven't listed before. You can do that for the real numbers, by keeping a set of all the unique real numbers you’ve produced before, then adding one random real number at a time, always checking that is not in the set already. And if you could go forever, then the probability of generating every real number is 1. However, infinity is unknowable. You can’t just say that the “infinite” cardinality of a set can just be reduced to a symbol, that is like saying you finished doing the calculation and that you know there’s a final result. But there isn’t a final result. The arguments in your explanation only prove that you cannot predict or choose an ordering for the real numbers before counting them. But if you produce random real numbers at each step and re-order them, you don’t need a predefined order. In fact the order can just be the order in which the real number was generated, regardless of its value, then you don’t even need to reorder them. Edit: replying to reply to this by LegionMammal978 as the nesting level doesn’t let me reply under it Thank you for the engaging conversation. You found the issue: Cantors argument assumes you can finish an infinite process and then add an extra step… Please see this other comment: https://news.ycombinator.com/item?id=35311403 https://news.ycombinator.com/item?id=35311403
    - LegionMammal978 4y ago
      > However, infinity is unknowable. You can’t just say that the “infinite” cardinality of a set can just be reduced to a symbol, that is like saying you finished doing the calculation and that you know there’s a final result. But there isn’t a final result. We put a symbol on cardinality because it lets us make statements about the properties of sets, like whether or not we're able to map them one-to-one with other sets. In turn, this often helps us with proving statements about finite objects. (Technically, they let us quantify over different scenarios: say what can't happen, or can happen, or might not happen, or must always happen.) To repeat my example, we say with symbols that x + 1 > x for all the infinitely many numbers x, even though we can't physically check all of them. But if you hand me some physical number X and I add 1 to it, I can apply the abstract statement to know that I'll get a bigger number in that scenario. > The arguments in your explanation only prove that you cannot predict or choose an ordering for the real numbers before counting them. But if you produce random real numbers at each step and re-order them, you don’t need a predefined order. In fact the order can just be the order in which the real number was generated, regardless of its value, then you don’t even need to reorder them. I don't understand what you're trying to say here. In more finite terms (given that random real numbers already contain infinite information), the diagonalization argument says, "You can keep generating individual real numbers as long as you want, by any process. But there'll always be at least one real number that you'll never ever count to, no matter how long you keep generating numbers." (You can't say the same thing about the natural numbers: some methods of counting them will, in fact, reach every number sooner or later. That's the crucial distinction.) This does not require any kind of "predefined order". In fact, the diagonalization has to listen to all the numbers we generate just to provide an example of an unreachable number. So at any given point in the process, we can't know in full what the unreachable number is (without knowing the generation process); we only know its initial digits. But the limit of these partial answers is the full unreachable number, which had already been there in the first place. (I hope you accept the idea of an arithmetic limit; without it, we'd have trouble with basic things like justifying that 0.999... = 1, or resolving Zeno's paradoxes of motion. It's an intrinsic part of what a real number is.)

by haneul 4y ago

nico 4y ago

It only strikes me that we are trying so hard to know the unknowable. Infinite essentially mean unknown. How could you possible compare the size of two unknowns? You just can’t, but we want to be able to reduce something infinite/unknowable, to a finite symbol that we can use in our limited language. Edit: for sibling comment -> how would you know the cardinality of a set without counting it? And if you have two infinite sets, how would you ever finish counting them to determine their cardinality and compare them? It doesn’t even make sense to talk about the cardinality of an infinite set, infinity is not a number. We pretend that the infinity symbol can somehow magically summarize and quantize something that is essentially unknowable. The infinity symbol doesn’t make something magically finite. The cardinality of an infinite set is unknowable. It is not “infinity”.

LegionMammal978 4y ago
> Infinite essentially mean unknown. How could you possible compare the size of two unknowns? The point is, that's not really what infinite means. A set being infinite just means that we can start listing elements, and we'll always be able to find a new element that we haven't listed before. To compare two infinite sets A and B, we can consider functions F that take any element from A and output some element from B. If A and B have the same cardinality, then we can always describe a certain function F that can potentially output any element from B, if we know the right input from A. But if B has a greater cardinality than A, then we cannot find any such function. No matter which function F we choose, there will always be certain values in B that can never be output by F given any input in A. Thus, we say that B has "more elements" than A in a certain sense. In the case of Cantor's argument, A is the set of natural numbers, B is the set of real numbers, and F is the ordering we choose. A variation of Cantor's argument can be expressed fully in terms of finite objects. Suppose we have a function F such that F(i) = f_i, where f_i is a function such that f_i(j) represents the jth digit of the ith real number in the sequence. Then, we can write a new function g(j) = (F(j)(j) + 2) mod 10, or some variation. This function g(j) similarly represents a real number. Now, we can take any i we want and start comparing f_i(1) vs. g(1), f_i(2) vs. g(2), etc. After a finite amount of time, we'll always reach a point where the two functions differ. This means that the two functions represent two different real numbers. Therefore, the real number represented by g is not represented by any of the f_i, no matter which function F we start with.
- nico 4y ago
  Thank you, I really appreciate your thoughtful explanation. > A set being infinite just means that we can start listing elements, and we'll always be able to find a new element that we haven't listed before. You can do that for the real numbers, by keeping a set of all the unique real numbers you’ve produced before, then adding one random real number at a time, always checking that is not in the set already. And if you could go forever, then the probability of generating every real number is 1. However, infinity is unknowable. You can’t just say that the “infinite” cardinality of a set can just be reduced to a symbol, that is like saying you finished doing the calculation and that you know there’s a final result. But there isn’t a final result. The arguments in your explanation only prove that you cannot predict or choose an ordering for the real numbers before counting them. But if you produce random real numbers at each step and re-order them, you don’t need a predefined order. In fact the order can just be the order in which the real number was generated, regardless of its value, then you don’t even need to reorder them. Edit: replying to reply to this by LegionMammal978 as the nesting level doesn’t let me reply under it Thank you for the engaging conversation. You found the issue: Cantors argument assumes you can finish an infinite process and then add an extra step… Please see this other comment: https://news.ycombinator.com/item?id=35311403 https://news.ycombinator.com/item?id=35311403
  - LegionMammal978 4y ago
    > However, infinity is unknowable. You can’t just say that the “infinite” cardinality of a set can just be reduced to a symbol, that is like saying you finished doing the calculation and that you know there’s a final result. But there isn’t a final result. We put a symbol on cardinality because it lets us make statements about the properties of sets, like whether or not we're able to map them one-to-one with other sets. In turn, this often helps us with proving statements about finite objects. (Technically, they let us quantify over different scenarios: say what can't happen, or can happen, or might not happen, or must always happen.) To repeat my example, we say with symbols that x + 1 > x for all the infinitely many numbers x, even though we can't physically check all of them. But if you hand me some physical number X and I add 1 to it, I can apply the abstract statement to know that I'll get a bigger number in that scenario. > The arguments in your explanation only prove that you cannot predict or choose an ordering for the real numbers before counting them. But if you produce random real numbers at each step and re-order them, you don’t need a predefined order. In fact the order can just be the order in which the real number was generated, regardless of its value, then you don’t even need to reorder them. I don't understand what you're trying to say here. In more finite terms (given that random real numbers already contain infinite information), the diagonalization argument says, "You can keep generating individual real numbers as long as you want, by any process. But there'll always be at least one real number that you'll never ever count to, no matter how long you keep generating numbers." (You can't say the same thing about the natural numbers: some methods of counting them will, in fact, reach every number sooner or later. That's the crucial distinction.) This does not require any kind of "predefined order". In fact, the diagonalization has to listen to all the numbers we generate just to provide an example of an unreachable number. So at any given point in the process, we can't know in full what the unreachable number is (without knowing the generation process); we only know its initial digits. But the limit of these partial answers is the full unreachable number, which had already been there in the first place. (I hope you accept the idea of an arithmetic limit; without it, we'd have trouble with basic things like justifying that 0.999... = 1, or resolving Zeno's paradoxes of motion. It's an intrinsic part of what a real number is.)