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Comment by szemet | Hacker News Reader

Any distribution is good with these properties, e.g. just use standard normal distribution, it is available in every statistics libraries, but easy to generate from uniform random reals, for example:

https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform

Also wikipedia has an easy to understand summary too, in the two envelope problem article:

https://en.wikipedia.org/wiki/Two_envelopes_problem#Randomiz...

Here it explains with numbers larger than 0, and uses exponential distribution (even easier to generate from uniform random variable).

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Almost every number chosen randomly between negative infinity and infinity would have so many digits that it would not fit on any computer currently in existence. It would not fit on a piece of paper. In the original problem, a number is written down on one piece of paper. That limits the possible numbers involved considerably.

If instead, the game was played between two supernatural beings with access to an inifinite piece of paper, and infinite time to write the numbers down, that would be a very different game.

This demonstrates that a sequence of coin flips which could be carried out in the lifetime of a human, or by a computer while the universe exists, converted to a number between 0 and 1, and then mapped to the whole number line is not random between negative infinity and positive infinity. Because of the small number of coin flips, you will never end up with any of the really big numbers. And the really big numbers is where it's at.

Generally mathematical theorems involving real numbers, do not care about computability (except if you are a finitist https://en.m.wikipedia.org/wiki/Finitism ) ;)

On the other hand my intuition is: flipping a coin (generating next digit of a real number between 0 and 1) and then trasforming it to e.g. an exponential random variable should always take you to the point where you can decide about the ordering in countably finite coin toss, with probability one...

Mathematicians don't deserve the right to amaze us with counterintuitive phenomena that can only exist in abstract theory outside the physical universe. All that shows is that their mathematical structures aren't faithful to the physical reality they pretend to model.

A Turing machine runs on a theoretically infinitely long tape in both directions. That doesn't make the concept of a Turing machine useless.

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