Timeline for Is there an "anti-choice axiom"?
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| Feb 14, 2022 at 7:38 | comment | added | Asaf Karagila♦ | @AlecRhea: I have said it elsewhere before, I am not one for visualizing to begin with. Maybe that's why set theory was always more attractive that other subjects. I do have some intuitive grasp, but the situation is so diverse without choice, so it's not as honed as the "at home intuition" you'd might think about. | |
| Feb 14, 2022 at 1:21 | comment | added | Alec Rhea | Has working with all of these very strange models of ZF eventually given you an intuition for how to 'imagine' sets like this, in similar fashion to how people wonder about Thurston's ability to visualize the fourth dimension after working for so long with manifolds? Obviously you have developed very sophisticated machinery that allows you to technically construct these strange members of the multiverse, but I'm curious if you 'feel at home' in them after working with them for so long in the same way we canonically 'feel at home' in ZFC. | |
| Feb 13, 2022 at 17:06 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
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| Feb 13, 2022 at 17:06 | comment | added | Asaf Karagila♦ | @Acccumulation: Thanks, that is a typo. And yes, of course you find it hard to see how that's possible. It's just plain weird. Even sets like $\Bbb R$. | |
| Feb 13, 2022 at 16:23 | comment | added | Acccumulation | "A admits a choice function is ∏A∖{∅} is non-empty." is -> if? And I have trouble seeing how an infinite set can be the union of two sets of strictly smaller cardinality. | |
| Feb 12, 2022 at 19:24 | comment | added | Asaf Karagila♦ | Seems like something I addressed in arxiv.org/abs/1911.09285 | |
| Feb 12, 2022 at 18:46 | comment | added | Gro-Tsen | How about an axiom with the following flavor: “for every infinite set $I$ and any $I$-indexed family $(G_i)_{i\in I}$ of groups, there exists an $I$-indexed family $(X_i, \sigma_i)_{i\in I}$ of pairs such that $X_i$ is a set, $\sigma_i \colon G_i \times X_i \to X_i$ is an action of $G_i$ on $X_i$ making it into a principal homogeneous set (i.e., $X_i\neq 0$ and $G_i \to X_i, g\mapsto \sigma_i(g,x)$ is bijective for all $x\in X_i$), and $\prod_{i\in I} X_i = \varnothing$”. Might this, or some variation around it, be consistent and a reasonable “anti-choice” axiom? | |
| Feb 12, 2022 at 15:00 | history | answered | Asaf Karagila♦ | CC BY-SA 4.0 |