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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 t

by 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.
- haneul 4y ago
  I’m not trying to make infinity finite. I am just giving the infinity of the positive integers a computationally verifiable property. > If you generate random reals at each step, you can generate a sequence as long as you want, without ever “getting stuck”. You don’t understand your own argument. At no point did you ever generate an irrational number, nor can you. You can only generate approximations of them, all of which are rational. You are trying to make infinity finite by reducing it to “generate a random real”. As it so happens, the infinity of the rationals is the same as the infinity of the positive integers, and demonstrably so. But the reals are not only the rationals. Or, you could say the reals do not exist. Only rational numbers do.
  - nico 4y ago
    > At no point did you ever generate an irrational number, nor can you. You can only generate approximations of them, all of which are rational. If you can’t generate an irrational number, then you can never have a set that contains it. Again, you want to use a finite label “irrational”, to denote the idea of an infinite process (generating some irrational number one digit at a time without stopping). You cannot count an infinite process. It doesn’t matter if it’s a rational number, the reals or the naturals.
    - haneul 4y ago
      The thing you are failing to understand is that your statement of “generate a random real” at each step is meaningless because each such step cannot be guaranteed to terminate. Which means the generator function you have been espousing this entire time is in fact just as much symbolic fiction as the symbol of pi. Consider the while loop, while (1 === 1) { print “Step {step++}” } Though clearly an infinite process, each step within the loop clearly terminates and has a loop index. That is, we will see “Step 100” printed, “Step 9999999”, etc given enough time. In a sense, countable, for a particular definition of countable. On the other hand, while (1 === 1) { print “Step {step++}“ random_real() } This, your favored generator, is not guaranteed to ever print “Step 2”, because at step 1, it might try to print pi, which is a fiction that will never terminate and become real. Thus not countable. See the difference?

by haneul 4y ago

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.

haneul 4y ago
I’m not trying to make infinity finite. I am just giving the infinity of the positive integers a computationally verifiable property. > If you generate random reals at each step, you can generate a sequence as long as you want, without ever “getting stuck”. You don’t understand your own argument. At no point did you ever generate an irrational number, nor can you. You can only generate approximations of them, all of which are rational. You are trying to make infinity finite by reducing it to “generate a random real”. As it so happens, the infinity of the rationals is the same as the infinity of the positive integers, and demonstrably so. But the reals are not only the rationals. Or, you could say the reals do not exist. Only rational numbers do.
- nico 4y ago
  > At no point did you ever generate an irrational number, nor can you. You can only generate approximations of them, all of which are rational. If you can’t generate an irrational number, then you can never have a set that contains it. Again, you want to use a finite label “irrational”, to denote the idea of an infinite process (generating some irrational number one digit at a time without stopping). You cannot count an infinite process. It doesn’t matter if it’s a rational number, the reals or the naturals.
  - haneul 4y ago
    The thing you are failing to understand is that your statement of “generate a random real” at each step is meaningless because each such step cannot be guaranteed to terminate. Which means the generator function you have been espousing this entire time is in fact just as much symbolic fiction as the symbol of pi. Consider the while loop, while (1 === 1) { print “Step {step++}” } Though clearly an infinite process, each step within the loop clearly terminates and has a loop index. That is, we will see “Step 100” printed, “Step 9999999”, etc given enough time. In a sense, countable, for a particular definition of countable. On the other hand, while (1 === 1) { print “Step {step++}“ random_real() } This, your favored generator, is not guaranteed to ever print “Step 2”, because at step 1, it might try to print pi, which is a fiction that will never terminate and become real. Thus not countable. See the difference?