Timeline for Axiom of determinacy as setting for studying rigs with $\operatorname{Aut}(\mathcal{M})\cong\operatorname {Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$?
Current License: CC BY-SA 4.0
19 events
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| Jan 18, 2021 at 0:39 | comment | added | Noah Schweber | @SylvainJULIEN Keep in mind that all concrete results gotten will only depend on the large cardinal hypotheses, not on determinacy per se. Specifically, by Shoenfield absoluteness all $\Pi^1_2$ sentences are absolute between $V$ and $L(\mathbb{R})$; if there are sufficient large cardinals in $V$, then $L(\mathbb{R})\models\mathsf{ZF+DC+AD}$, so if we can decide a $\Pi^1_2$ sentence from $\mathsf{ZF+DC+AD}$ then we can decide it the same way from $\mathsf{ZFC+}$ [large cardinals]. And for example the Riemann hypothesis is ($\mathsf{ZF}$-provably equivalent to a sentence which is) $\Pi^0_1$. | |
| Jan 18, 2021 at 0:28 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Jan 18, 2021 at 0:14 | comment | added | Sylvain JULIEN | I planned to write out details of how to extend morphisms of L-rigs into morphisms of fields today through categories with the friend I talked about above, but I couldn't see her because of covid-19 :-(. If you read French, I can send you the pdf we would have worked with. | |
| Jan 18, 2021 at 0:05 | comment | added | Wojowu | Thanks for clarifying about the rig structure, that was not clear to me, as well as for deconvoluting the question. It is still not clear to me how you wish to use AD anywhere. The isometry groups of zero sets you mention are almost always going to be trivial or consist of two elements. This is provably (in ZF) true for Riemann zeta. While I don't know how your proposed relation with $Gal(\overline\mathbb Q/\mathbb Q)$ works, it is also not clear to be where AC or AD would come in - as $\overline\mathbb Q$ is a countable field, nearly all its properties are derivable in ZF too. | |
| Jan 17, 2021 at 23:49 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Jan 17, 2021 at 22:48 | comment | added | user44143 | I might title this as: "Would the axiom of determinacy provide a good setting for studying rigs with $\operatorname{Aut}(\mathcal{M})\simeq\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$? The current title seems to promise a closer relationship with RH than is justified or explained by the body. | |
| Jan 17, 2021 at 22:46 | review | Close votes | |||
| Jan 24, 2021 at 3:07 | |||||
| Jan 17, 2021 at 22:44 | comment | added | Wojowu | @Rahman.M I'm not sure I understand your comment. Con(ZFC) is also (equivalent to) a statement about natural numbers, but it is implied by AD (taking ZF as our metatheory) | |
| Jan 17, 2021 at 22:32 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Jan 17, 2021 at 22:31 | comment | added | Sylvain JULIEN | I tried to "deconvolute" my phrasing, hope it's better now. | |
| Jan 17, 2021 at 22:30 | comment | added | Sylvain JULIEN | The laws of composition of L-rigs are the usual product and the Rankin-Selberg convolution, no "sum" is involved. | |
| Jan 17, 2021 at 22:29 | comment | added | Wojowu | @Rahman.M Although ZF+AC cannot prove any arithmetic statements which ZF can't, the same is not true of ZF+AD (so in principle it could be the case ZF+AD proves RH but ZF does not). It is not clear to me how Sylvain proposes to use AD though. If he only wishes to use that C has no discontinuous automorphisms, we can use a weaker hypothesis (all sets of reals have property of Baire), which I believe does not add any arithmetic consequences to ZF. | |
| Jan 17, 2021 at 22:28 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Jan 17, 2021 at 22:26 | comment | added | Wojowu | Is the L-rig (generated by?) powers of Riemann zeta even an L-rig? I don't think sums of those powers are necessarily L-functions (in the sense of Selberg class at least). | |
| Jan 17, 2021 at 22:23 | comment | added | Sylvain JULIEN | Also note that for the "ordinary" RH only the most simple L-rig is needed, as it consists of powers of the Riemann zeta function. | |
| Jan 17, 2021 at 22:21 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Jan 17, 2021 at 22:21 | comment | added | Sylvain JULIEN | Determinacy, indeed. A friend of mine also told me I tended to use too much natural language in my questions, and I have a hard time improving this. I'll try to though. | |
| Jan 17, 2021 at 22:16 | comment | added | Wojowu | Almost whole second paragraph is composed of a single sentence which is so convoluted I'm having hard time understanding what it is trying to say. In particular what's unclear to me is what axiom of determinacy has to do with everything else. One question I have is, without axiom of choice, can you prove a maximal L-rig even exists? I am not aware of a construction, and Zorn's lemma is not available without choice. | |
| Jan 17, 2021 at 21:54 | history | asked | Sylvain JULIEN | CC BY-SA 4.0 |