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I had seen the posting "https://math.stackexchange.com/questions/29381/picking-from-an-uncountable-set-axiom-of-choice?rq=1" but I'm still curious about it especially on the matter of the manner that pick arbitrary real number. If I declare that the manner of the picking real number as like this: Let me pick number from [0,1]-It is widely known that R has same cardinality with it- first, I choose first digit number from the set {1,2,3,4,5,6,7,8,9,0} then likewise second too. then It requires 'infinite choosing' from the set to choose arbitrary real number in this point of view, I wanna know that this Pov has some error.

I have one possibable answer that the infinite choosing of same set is not the case of Axiom of choice. but I have no certainty about it.

Nuga 1399
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  • A real number is not its decimal expansion. When you choose a number, you're not choosing it "once digit at a time". Choosing a real number is choosing a single element from a non-empty set. Full stop. – Asaf Karagila May 07 '23 at 17:41

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