Timeline for How much of mathematical General Relativity depends on the Axiom of Choice?
Current License: CC BY-SA 4.0
14 events
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| Aug 22, 2023 at 22:03 | comment | added | Asaf Karagila♦ | @TimothyChow: Having said that, it is of course not necessarily the case for general metric spaces (or rather, ordered metric spaces, where Dedekind completions make sense). | |
| Aug 22, 2023 at 12:10 | comment | added | Timothy Chow | @AsafKaragila Ah, it seems I got mixed up...thanks. | |
| Aug 22, 2023 at 6:56 | comment | added | Asaf Karagila♦ | @TimothyChow: You mean in the absence of LEM, I think. The Cauchy completion and Dedekind completion are equivalent in ZF. | |
| Aug 21, 2023 at 22:55 | comment | added | Timothy Chow | @Z.M See Section 3 of Ruitenberg, Constructing roots of polynomials over the complex numbers. See also Richman, Constructive mathematics without choice, who showed that if you drop both choice and excluded middle, then even the fundamental theorem of algebra has issues. | |
| Aug 21, 2023 at 22:20 | comment | added | Z. M | @TimothyChow What are Cauchy reals (without DC or CC)? I guess that it is the completion of the uniform space $\mathbb Q$, and do we need choice to identify with the Dedekind completion? | |
| Aug 21, 2023 at 20:25 | comment | added | Timothy Chow | @Z.M In the absence of choice, Dedekind reals and Cauchy reals are not the same. The traditional development of measure theory uses choice. I'm not sure if these count as "first year analysis" though. | |
| Aug 21, 2023 at 16:47 | comment | added | Z. M | @KevinCasto Do you have any concrete example in first-year analysis where the DC or CC is essential, beyond sequential criteria for limits or sequential compactness? I do not remember the importance of "reduction to sequences" there. | |
| Aug 21, 2023 at 16:34 | vote | accept | Pelota | ||
| Aug 21, 2023 at 12:07 | comment | added | Timothy Chow | See Martin Väth's answer to another MO question for more information about how much choice is needed for basic analysis. | |
| Aug 21, 2023 at 11:06 | history | edited | LSpice | CC BY-SA 4.0 |
Fixing my wrong edit, sorry
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| Aug 21, 2023 at 10:35 | history | edited | Igor Khavkine | CC BY-SA 4.0 |
added 6 characters in body
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| Aug 21, 2023 at 7:19 | comment | added | Kevin Casto | I think it's important here to separate countable or dependent choice from full AC. Even lots of first year analysis depends, at least as written, on countable choice. Personally I think it's fine to take CC or even DC as simply "true", but carefully track any stronger uses of choice, which are extremely hard to imagine being physical. | |
| Aug 21, 2023 at 2:29 | history | edited | LSpice | CC BY-SA 4.0 |
Names of links
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| Aug 21, 2023 at 0:45 | history | answered | Igor Khavkine | CC BY-SA 4.0 |