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I am aware that the following question/discussion is ill-defined, and perhaps has more to do with philosophy than mathematics.

My brother and I were talking and the topic came to “picking a random natural number.” I told him that to my best knowledge, as well as to my logic, there is no such thing as picking a random natural number without setting an upper limit, such as picking a random natural number from the set of natural numbers. Further, I added that one cannot write a computer program that would pick a random natural number from the set of natural numbers. (For the sake of simplicity, the process of how “randomness” is determined, or whether perfect randomness can be achieved is out of discussion.)

Then, the topic came to the probability of Person A and Person B picking the same natural number. And here is how I managed to confuse myself: Suppose A and B think of a natural number in any way they wish; they are even allowed to invent their own notations for very large numbers. What's the probability of A's choice being equal to B's choice?

Argument 1: The answer is not zero, because a human's capacity of picking a random natural number is not infinitely large.

Argument 2: The answer is lower than any given ratio, because one cannot set an upper limit to human imagination. (Whatever the limit, I say limit plus one, etc.) (Yet, human imagination is not infinite.)

My question: Is this discussion meaningful in any way? Does it belong to mathematics? Could it ever be possible to simulate the above discussion via a computer program? (That, is it theoretically possible to write a computer program such that one can never know the upper limit the program will output?)

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What is not possible is to pick a random natural number in a way such that every natural number is equally likely.

If you don't require every number to have the same probability, then one can certainly come up with a well-defined probabilistic experiment where every natural is a possible outcome -- for example, keep flipping a fair coin until you get tails, and count how many heads you got before this happened.

Then, the topic came to the probability of Person A and Person B picking the same natural number. And here is how I managed to confuse myself: Suppose A and B think of a natural number in any way they wish; they are even allowed to invent their own notations for very large numbers

"Any way they wish" does not produce definite probabilities, even if there are finitely many options. You would have to start by assuming some probabilities for how each player selects a strategy for picking his number first -- otherwise you simply don't have enough data to bring probability theory to bear on the situation.

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This would be an interesting experiment in psychology or human behavior. What is the distribution of natural numbers "randomly" picked by volunteers? I suspect you'll see a bell-like distribution with a long tail where the 1st standard deviation is < 1000. Once you have that curve defined with experimental data, real probabilities can be calculated.

With a nod to the Sapir–Whorf hypothesis, where words limit thought, most people are unfamiliar with number names above a trillion. Is anyone likely to guess 103 ‎quadrillion? So assuming a max of 999,999,999,999,999 + 1 for smart alecs that know the value of a google, my guess for a lower limit of the probability is $1$ in $10^{15}$ but would not be surprised if experiments found it as high as $1$ in $10^4$. It would depend on how the question was phrased and if volunteers were encouraged to think of "big" numbers.

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