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Tim Hosgood (Oct 11 2021 at 10:11):Andreas Blaas: A topos view of axioms of choice for finite sets
Tarski proved (in set theory without choice) that if one assumes that all families of 2-element sets have choice functions then one can prove that all families of 4-element sets have choice functions. Mostowski (1937) investigated similar implications, giving number-theoretic and group-theoretic conditions, some necessary and some sufficient for such implications. But some questions remained unsolved, in particular: Do choice from 3-element sets, from 5-element sets, and from 13-element sets together imply choice from 15-element sets. Gauntt (1970) resolved those remaining questions, using group-theoretic criteria. I plan to describe some of this work and to explain what it has to do with topos theory.
Tim Hosgood (Oct 11 2021 at 10:11):this Thursday, back to the usual time of 17:00 UTC
Zoom: https://topos-institute.zoom.us/j/5344862882?pwd=Znh3UlUrek41T3RLQXJVRVNkM3Ewdz09
YouTube: https://www.youtube.com/watch?v=Y-Pfh9VJKBc
John Baez (Oct 11 2021 at 16:07):Do choice from 3-element sets, from 5-element sets, and from 13-element sets together imply choice from 15-element sets.
Crazy - but fascinating!
Tim Hosgood (Oct 14 2021 at 16:44):starting in 15 minutes!
James Deikun (Oct 14 2021 at 18:11):Great talk, and it points to a place where topos theory really illuminates something fundamental about conventional set theory.
I think a way to interpret things geometrically is 'if we had a device to take square roots of complex numbers, we could also use it to take fourth roots of complex numbers'. And from the combinatorial proof we could probably extract an algorithm for this.
Antonin Delpeuch (Nov 22 2021 at 17:59):It was a really nice talk! I just watched it now and thoroughly enjoyed the story. Of course I got lost towards the end, but that is normal. Andreas can probably not read this but he must be a great teacher, the exposition was very neat.