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If you recall, your original comment said “ If you say one infinity is larger/smaller than another, you’d be saying that the complete map for one is larger/smal

by haneul 4y ago

If you recall, your original comment said “ If you say one infinity is larger/smaller than another, you’d be saying that the complete map for one is larger/smaller than for the other one. But if they are both infinite, you will never stop counting, so you just can’t know.”. So, if I can map all of the positive integers onto the reals and provably have reals that cannot be generated, that should suffice, no? Take the reals from 0 to 1. For each positive integer X, flip the digits and put it behind the decimal. So, 437 gets mapped to 0.734, 1000 gets mapped to .0001, etc. None of the transcendental numbers between 0 and 1 will get mapped onto. This is because all positive integers are a finite series of digits, whereas all transcendental numbers provably do not terminate. It is not possible to flip a finite sequence of digits and get a non terminating sequence of digits. So, now you have a generator function that can go through the positive integers one by one and never generate any members of an entire class of real numbers. FYI I find this quite useful in clarifying my own thinking so feel free to tell me why that isn’t convincing to you.

nico 4y ago
But you are imagining “all the numbers between 0 and 1”, those numbers don’t/won’t exist until you generate them. You cannot perform the action of mapping. You can give me a formula to start mapping. But I would never finish. So you are imagining the result. There is no result, the mapping is a process, not a finite result that you can compare to another finite result. As long as you keep counting (producing naturals one after another), I can keep producing random real numbers and sorting them. There’s no point at which either one of us cannot generate one additional number. And if we keep going to infinity, then the random generator will have generated all possible real numbers.
- haneul 4y ago
  Does pi/4 not exist? If I draw a circle of diameter 1 and erase all but one quadrant, I now have constructed a segment of length pi/4. This happens to be a transcendental number between 0 and 1. Stop using your random number generator. It is making you miss the point. It has no special value. For some reason you are assigning it a special value the same way people might assign numerical greater than order a special value. If you use the flipped positive integer generator instead, it is quite obvious that you will never generate pi/4 no matter how long you run that generator, yet that generator will run through all possible positive integers.
nico 4y ago
> Does pi/4 not exist? If I draw a circle of diameter 1 and erase all but one quadrant, I now have constructed a segment of length pi/4. This happens to be a transcendental number between 0 and 1. Pi/4 the symbol exists. Pi/4 the calculated number output doesn’t exist. You can’t finish calculating it. And if you calculated a finite approximation, then all you’d be doing is just connect one symbol (Pi/4) to another symbol (output of your calculation). You keep thinking hat assigning symbols/labels to infinity somehow gives you an instant result of an infinite process/computation. That’s the problem.
- haneul 4y ago
  You don’t need to finish calculating pi/4 to know that it is a non-computable number, whereas all positive integers are computable, which is why the length of that segment will never show up in the generator. My assertion is simply that the infinity of the positive integers is different from the infinity of the reals in the sense that the infinity of the positive integers can be computed to arbitrarily large measure given enough time. Whereas for the reals, we’ll forever be stuck on pi/4 and never get to e. That given arbitrary time, we can only compute a non-measurable number of reals. Which makes the infinity of the positive integers more “real” than the infinity of the reals, because we can use measurable things in algorithms, whereas non-measurable things require a leap of imaginative voodoo that I imagine you would rather not take. If you prefer to take the infinity of the positive integers as literally the same as the infinity of the reals, then you’d have to believe that all finite sets of reals have positive measure within the reals. Sure, both infinities can be considered “unknowable” in that we cannot generate the entire thing so cannot generate/know all their properties, but we can generate the measure property. So they should both satisfy it. Also, intuitively I think we should be able to prove the halting problem wrong, if all finite sets of reals are positively measurable, but don’t quote me on that! But I feel like taking such a property would imply that the spectral gap problem is computable, which then bunks the halting problem, and the concept of non-computable numbers altogether.
  - nico 4y ago
    > My assertion is simply that the infinity of the positive integers is different from the infinity of the reals in the sense that the infinity of the positive integers can be computed to arbitrarily large measure given enough time. Again the same issue. You are trying to make infinity finite by reducing it to “arbitrarily large measure”. > Whereas for the reals, we’ll forever be stuck on pi/4 and never get to e. That given arbitrary time, we can only compute a non-measurable number of reals. If you generate random reals at each step, you can generate a sequence as long as you want, without ever “getting stuck”. You are choosing to get stuck.

by haneul 4y ago

nico 4y ago

But you are imagining “all the numbers between 0 and 1”, those numbers don’t/won’t exist until you generate them. You cannot perform the action of mapping. You can give me a formula to start mapping. But I would never finish. So you are imagining the result. There is no result, the mapping is a process, not a finite result that you can compare to another finite result. As long as you keep counting (producing naturals one after another), I can keep producing random real numbers and sorting them. There’s no point at which either one of us cannot generate one additional number. And if we keep going to infinity, then the random generator will have generated all possible real numbers.

haneul 4y ago
Does pi/4 not exist? If I draw a circle of diameter 1 and erase all but one quadrant, I now have constructed a segment of length pi/4. This happens to be a transcendental number between 0 and 1. Stop using your random number generator. It is making you miss the point. It has no special value. For some reason you are assigning it a special value the same way people might assign numerical greater than order a special value. If you use the flipped positive integer generator instead, it is quite obvious that you will never generate pi/4 no matter how long you run that generator, yet that generator will run through all possible positive integers.

nico 4y ago

> Does pi/4 not exist? If I draw a circle of diameter 1 and erase all but one quadrant, I now have constructed a segment of length pi/4. This happens to be a transcendental number between 0 and 1. Pi/4 the symbol exists. Pi/4 the calculated number output doesn’t exist. You can’t finish calculating it. And if you calculated a finite approximation, then all you’d be doing is just connect one symbol (Pi/4) to another symbol (output of your calculation). You keep thinking hat assigning symbols/labels to infinity somehow gives you an instant result of an infinite process/computation. That’s the problem.

haneul 4y ago
You don’t need to finish calculating pi/4 to know that it is a non-computable number, whereas all positive integers are computable, which is why the length of that segment will never show up in the generator. My assertion is simply that the infinity of the positive integers is different from the infinity of the reals in the sense that the infinity of the positive integers can be computed to arbitrarily large measure given enough time. Whereas for the reals, we’ll forever be stuck on pi/4 and never get to e. That given arbitrary time, we can only compute a non-measurable number of reals. Which makes the infinity of the positive integers more “real” than the infinity of the reals, because we can use measurable things in algorithms, whereas non-measurable things require a leap of imaginative voodoo that I imagine you would rather not take. If you prefer to take the infinity of the positive integers as literally the same as the infinity of the reals, then you’d have to believe that all finite sets of reals have positive measure within the reals. Sure, both infinities can be considered “unknowable” in that we cannot generate the entire thing so cannot generate/know all their properties, but we can generate the measure property. So they should both satisfy it. Also, intuitively I think we should be able to prove the halting problem wrong, if all finite sets of reals are positively measurable, but don’t quote me on that! But I feel like taking such a property would imply that the spectral gap problem is computable, which then bunks the halting problem, and the concept of non-computable numbers altogether.
- nico 4y ago
  > My assertion is simply that the infinity of the positive integers is different from the infinity of the reals in the sense that the infinity of the positive integers can be computed to arbitrarily large measure given enough time. Again the same issue. You are trying to make infinity finite by reducing it to “arbitrarily large measure”. > Whereas for the reals, we’ll forever be stuck on pi/4 and never get to e. That given arbitrary time, we can only compute a non-measurable number of reals. If you generate random reals at each step, you can generate a sequence as long as you want, without ever “getting stuck”. You are choosing to get stuck.