In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see:

https://en.wikipedia.org/wiki/Almost_surely

This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like:

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Measure theory deals with resolving this:

https://en.wikipedia.org/wiki/Measure_(mathematics)

What's an anagram for Banach Tarski?

Banach Tarski Banach Tarski.

Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.

Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense.

In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained by finite bit representations of numbers.

> In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer

An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.

A Turing machine can't pick random numbers of any kind.

Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.

What about PI? We can represent it in terms of "we know what we are talking about" and we can distinguish it from other numbers.

You can only have countable number of first or second order logic statements each defining a specific real number.

Those things sound equivalent, but I don't want to be the one to try to prove it.

Run a random number generator (0-9) for each decimal position after the dot in parallel. This should do a trick.

*entirely different