Timeline for How much of the axiom of choice do you need in mathematics?
Current License: CC BY-SA 4.0
10 events
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| Aug 25, 2023 at 19:29 | history | edited | Alec Rhea | CC BY-SA 4.0 |
typo
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| Aug 15, 2023 at 16:36 | comment | added | Z. M | When you have a sheaf, either on a manifold, or "globally" on the category of all manifolds with some topology, it is zero if and only it is zero locally. In other words, there are two steps: 1. showing that it is a sheaf; 2. showing that it is locally zero (which is a computation in $\mathbb R^d$. The reduction to valuation rings is similar: 1. showing that is is a v-sheaf or arc-sheaf; 2. showing that it is zero on valuation rings (where the computation is much easier). We can also replace being zero by other properties, cf. Bhatt–Mathew, Cor 3.18,19&25. | |
| Aug 15, 2023 at 16:11 | comment | added | Mikhail Katz | @Z.M , thanks for this. Well, I work in differential geometry, but what do you mean by reducing general manifolds to open subsets of Euclidean space? Certainly locally they are, but not globally in general. | |
| Aug 15, 2023 at 16:07 | comment | added | Z. M | I am not technically equipped to extract a concrete "basic" problem, but if I understand correctly, it is basically used to reduce general rings to valuation rings (e.g. ultraproducts of valuation rings are still valuation rings). It is analogous to reducing general manifolds to open subsets of $\mathbb R^d$ in differential geometry. Hansen–Scholze gives a construction of perverse $t$-structure using these techniques. | |
| Aug 15, 2023 at 13:35 | comment | added | Mikhail Katz | @Z.M : I tried reading Bhatt-Mathew but found the terminology rather inaccessible. Could you perhaps point out a basic problem in algebraic geometry that is usually solved using ultraproducts, without using hifalutin' language? I could then try to check whether this can also be done without ultrafilters along the lines of Blass's comment. | |
| Aug 13, 2023 at 22:06 | comment | added | Z. M | Ultrafilters and ultraproducts appear frequently in algebraic geometry, cf. §3.2 of Bhatt–Mathew. | |
| Aug 13, 2023 at 16:32 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Aug 13, 2023 at 14:11 | comment | added | Mikhail Katz | @SamSanders, Nelson's REPT turns out to be a subsystem of SCOT, which is conservative over ZF+ADC. | |
| Aug 13, 2023 at 14:08 | comment | added | Sam Sanders | One should perhaps mention Nelson's "radically elementary probability theory", which is based on very modest means (as opposed to the treatment not involving infinitesimals). | |
| Aug 13, 2023 at 13:11 | history | answered | Mikhail Katz | CC BY-SA 4.0 |