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Is it fundamentally wrong to think if the Axiom of Choice (AC) as a logical codification of indeterminism? Could it not be related to matching distributions with a random number generator?

Ekneumann
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  • I don't think so, although I'd like to hear more about how you are tying the concepts together. For me, AC essentially formulates that making arbitrary choices without a common structure in an infinite collection of sets is not possible in a finite amount of time. In other words, making a choice does take some time. – Rushabh Mehta Mar 03 '21 at 01:03
  • @Don: This is not about time. Nor it is about space. It's about something bigger than both. It's about math. – Asaf Karagila Mar 03 '21 at 01:14
  • @AsafKaragila I know, this is just my intuition on the axiom. – Rushabh Mehta Mar 03 '21 at 01:17
  • @Don: Yes, but that's the sort of thing that leads inexperienced people to think that the axiom of choice is like a JavaScipt (or whatever language) approach to iterators (e.g. forEach or map on an array). Which is very wrong. – Asaf Karagila Mar 03 '21 at 01:19
  • @AsafKaragila I disagree. I think it's precisely the opposite. I think AC says that as humans, the only way we can make arbitrary choices is like how an iterator does it. We can't, without AC, make an infinite set of choices all at once. – Rushabh Mehta Mar 03 '21 at 01:27

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Yes. It's wrong.

The axiom of choice is not choosing "random elements". That's what existential instantiation is doing, in some sense. The axiom of choice is best understood in some sense in a constructive framework: if you can choose an element from each set in a family of sets, then you're supposedly able to do it uniformly.

Of course, in classical mathematics this is not true, but the axiom of choice lets you overcome that issue.

The "non-constructive part" of the axiom of choice is really an artifact of existential instantiation, if you think about it, that's the part moving from "there is a choice function" to "$f$ is a choice function".

Asaf Karagila
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