Interesting! My experience is that scooping is less of an issue in math than in any of the science fields I have friends in. Papers are lower-stakes, there's less money involved, and if two of you are working on the same project you can just co-author.
(And if you have an independent paper, that can _also_ get published; your paper is distinct even if the result isn't. I think the PT HOMFLY polynomial was independently proven in like four different papers published within two years (and it's named so that all eight authors get credit).
But also, publication lags shouldn't lead to more scooping, because you can put it up on the arXiv at the beginning of the publication process, not the end. In my experience the paper is treated as "real" once it hits the arXiv; the acceptance is mostly a formality that lets us put it on our promotion packet.
But also, publication times don't lead to scooping generally because you
But what does "picking at random" mean? If you think it's obvious that you can just make infinitely many random choices at the same time, you're endorsing the axiom of choice.
(If you think you can make one random choice at a time, you're endorsing the axiom of finite choice, which is just true. Determinism doesn't matter here, but "at the same time" matters a lot.)
This is super cool and I wish I'd known about it before writing the piece; I'd have been comfortable being a lot more muscular in the conclusion. This seems like it makes precise my suggestion that the axiom of choice only matters for aggressively infinite and unphysical statements.
My take on this is that the axiom of choice allows you to produce infinite amounts of information, if and only if you start with infinite amounts of input.
Think about how much information is involved in presenting an infinite collection of sets! When I say even something as simple as "Let x be a real number" I'm already invoking infinitely many bits of information. Things have already gotten weird, you just haven't noticed because we've hidden it.
The axiom of choice is saying something like, okay, we somehow have this infinite pile of information sitting around; now we're allowed to interact with it. And if you don't like that, your problem might be with the infinite amount of information we started with. (Or it might not; yes-choice and no-choice are both totally valid positions.)
But in practice it doesn't matter that much because you never actually do start with infinite amounts of information, and so you don't need an infinite-amounts-of-information processor. But if we have finite amounts of information that we're pretending are infinite to simplify things, we can also process the finite amounts of information and pretend we're processing infinite information.
I'd agree with this entirely, honestly. (Including the sleight-of-hand I'm engaging in to make things seem exactly the right amount of weird; see footnote 14.)
Infinity is so weird that it's really hard to agree on what infinity "should" look like. And the axiom of choice only matters in situations where the weirdness of infinity really kicks in.
So yeah, some people look at the whole list of AoC equivalents and think they all seem pretty reasonable. And other people look at the list and think none of them seem that plausible. And a lot of us are split, and find some of the claims obviously true and others obviously false.
One of my goals in this piece was to try to let everyone see both sides of this: why you might find the axiom compelling and why you might find it troubling. And then to offer a pragmatic resolution at the end, which is basically another take on what you just said.
ketralnis is right that this only terminates if an algorithm exists, so the claim that this terminates is equivalent to the axiom of choice.
But I'd also add that you can have a choice function that isn't an "algorithm". An algorithm, at least in the sense I'd generally interpret the word, has finitely many instructions and at most countably many steps. If we have uncountably infinitely many uncountably infinite sets, it is possible to have a choice function that can't be described in a finite algorithm.
Like, think about a well-ordering of the reals. If you believe the axiom of choice, then one exists. But you can't tell me what it is, because that would involve handing me infinite amounts of information. And similarly you can't write down an algorithm to produce it, without writing down infinite amounts of data.
(And if you have an independent paper, that can _also_ get published; your paper is distinct even if the result isn't. I think the PT HOMFLY polynomial was independently proven in like four different papers published within two years (and it's named so that all eight authors get credit).
But also, publication lags shouldn't lead to more scooping, because you can put it up on the arXiv at the beginning of the publication process, not the end. In my experience the paper is treated as "real" once it hits the arXiv; the acceptance is mostly a formality that lets us put it on our promotion packet.
But also, publication times don't lead to scooping generally because you