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Let me describe the issue as follows. You have a real number lottery. Everyone gets to choose their own real number, say, and put it in a sealed envelope. You choose the winning number by whatever process you like. But, you must publish an actual number.FactChecker said:@PeroK , Suppose I define a selection process as follows:
I let you define a selection process on the [0,1] line segment that I have no knowledge of or influence on. Let ##P## denote the countable set of possible results of your process and ##I## denote the remainder of [0,1] of numbers that are impossible to select using your process. ##P## has measure zero and ##I## has measure 1. If I apply the Axiom of Choice to claim a chosen value ##c## from ##I##, I must say that it had probability zero, even though it was selected.
You are not allowed to say you picked "some" number ##c##, but you don't know what it is. Nor can you describe it in any way.
Then you are limited to the computable numbers.
It's nothing to do with the axiom of choice.