The discussion revolves around the concept of generating random integers with random lengths and whether this process can produce all possible integers. Participants explore the implications of such a generation method, particularly focusing on the probabilities involved and the nature of randomness in this context.
Participants do not reach a consensus; multiple competing views remain regarding the nature of randomness, the feasibility of generating all integers, and the implications of probability distributions.
Participants highlight limitations in defining how integers are selected at random, the implications of uniform versus non-uniform distributions, and the challenges of generating integers within finite time constraints.
First tell me how you would create that random number generator. If you want to create a random number in a finite time, you will only get random numbers in a finite range...serp777 said:Does the set of random integers with random lengths (the number of digits), which hypothetically would generate random numbers with random lengths for eternity, produce all possible integers? It seems to me that this is a natural conclusion but I've never seen a proof of this. A more incredible statement would be that there are integers which could not be produced via this process.
It is not possible to pick an integer at random such that all integers are possible and the probability distribution is uniform.micromass said:How do you even pick an integer at random? There is no way!
jbriggs444 said:It is not possible to pick an integer at random such that all integers are possible and the probability distribution is uniform.
My reading of the problem in #1 is that the requirement that the distribution be uniform has been relaxed but the requirement that the distribution have a non-zero probability for each and every integer has been retained. A procedure such as "flip a coin and count the number of tails prior to the first heads" is almost certain to work for this.