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If we first ask the question of choosing a random real number, say in the interval [0, 1], then one could come up with a similar process :

  • in 1/2 second, choose a random digit from 0 to 9
  • in 1/4 second, choose another random digit
  • and so on

After 1 second, by appending all the digits, we therefore would have created a random real number in the desired interval.

Obviously this doesn't completely answer this question as it still addresses the issue of producing an infinite amount of information in a finite amount of time, but let's just consider this as a possibility.

Every number obtained would be a non computable number in the sense that the only way to create it would be by the random process described above, which will never produce the same number twice.

But would it possible, to somehow extend this construction to generate a random integer ? In other words, given a random real r, generate a random integer from it, with every number having equal "probability" (not assigning a fixed positive probability to every number since this is impossible as the sum would be larger than 1, but such that the outcome would not favor any number over another) ?

If not, has this got something to do with the fact that the naturals are countable and the reals uncountable ?

Also, lets assume random number generation on the set of countably infinite integers is possible. What would the obtained sequence look like (i'm guessing it would have some form of fractal structure, with numbers getting ever bigger) ?

Asaf Karagila
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user403504
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    Your construction admits natural formalization as "uniform distribution on $[0, 1]$". But there is no such thing as "uniform distribution on natural numbers". What do you want to get in abstract sense, ignoring for a moment any process that should produce it? – mihaild Jan 17 '22 at 16:34
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    A worthwhile question is: what is the average number of digits in a uniformly randomly chosen integer? (Answer: it's infinite. This is a problem: no integer has infinitely many digits, but apparently "half" of uniformly randomly chosen integers do.) See also: https://math.stackexchange.com/questions/14777/why-isnt-there-a-uniform-probability-distribution-over-the-positive-real-number . Note that you want a process that produces a uniform distribution of (positive) integers, but no such distribution exists, so no such process exists. – Eric Towers Jan 17 '22 at 16:34
  • The odds of having a non-computable result of choosing a digit at random is indeed "almost surely". But it does not mean that it is impossible that you have always chosen 0. Indeed, any a posteriori sequences of choices is equally unlikely, regardless to the sequence itself. – Asaf Karagila Jan 17 '22 at 16:40
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    If this WOULD be possible, the probability to choose a number too large to be reasoanably described would be $1$. The same with random primes : If we COULD choose a random prime, the probability would be $1$ that it is not only an unknown prime, but so large that we would be unable to even grasp its magnitude. – Peter Jan 17 '22 at 16:55

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