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It is a theorem that adopting choice only retracts our universe. But it is intuitively obvious that choice only expands our universe. We are forced to conclude it does neither, and choice is in fact true.

WHat do you think? This is my philosophical argument for the axiom of choice.

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    Aren't you equivocating "choice?" The axiom of choice is not about having choices in the world.
    – user4894
    Sep 21 at 19:17
  • 2
    "It is a theorem that adopting choice only retracts our universe"??? And "what do you think" questions are off-topic here.
    – Conifold
    Sep 21 at 19:31
  • @Conifold I was #1 student at my university. Please don't ban me.
    – anon
    Sep 21 at 19:36
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    Improve the question so it makes more sense, and ask something more objectively answerable than "what do you think?" How axiom of choice is supposed to relate to the universe is obscure, and there is currently no intelligible argument in your post connecting them, so it is not even clear what you want users here to think about.
    – Conifold
    Sep 21 at 19:44
  • This question is worded so that it could mean almost anything. Sep 22 at 18:01
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This is just a string of unjustified claims.

"It is a theorem that adopting choice only retracts our universe"

While it is true that there are things that choice rules out (e.g. assuming choice there are no Reinhardt cardinals), there are also things that the negation of choice rules out (e.g. well-orderings of arbitrarily high levels of the cumulative hierarchy). This is true of any statement whatsoever, uninterestingly.

What you've written here is both unclear and very strong. What exactly does "retract our universe" mean, and in what sense is it true that choice does this and the negation of choice doesn't?

"But it is intuitively obvious that choice only expands our universe."

It is? Not to me. There are certainly authors who have argued for something like this statement (e.g. Maddy, if I recall correctly, argued that a "maximization" approach to the philosophy of mathematics motivates adopting choice), but I certainly wouldn't call this obvious even on days when I agree with it.

"We are forced to conclude it does neither"

I don't see why. In general when we have two conflicting intuitions the correct response is to further examine those intuitions, and this often leads us to rejecting one (or both!) of them - this is especially true when we have an intuition which seems to push against a theorem. And for that matter, I still have no idea what "retract our universe" means.

"and choice is in fact true."

This simply doesn't follow. In particular, everything above is totally symmetric between choice and the negation of choice: if we accept that choice "both should and shouldn't retract the universe" (whatever that means), then dually we accept that the negation of choice "both shouldn't and should retract the universe." OK, but then why doesn't this dual situation (which, again, I neither accept nor even parse) counts as evidence for the negation of choice exactly as much as it does for choice?

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A choice eliminates lots of possible outcomes.I'm not sure if that could be called retracting. They are not real things, only possibilities.

A choice adds information to the Universe, clarifies uncertainties. A choice says "this happens instead of those other possibilities". I'm not sure if that could be called "expanding".

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