Timeline for Open problem in analysis with just one quantifier?
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2021 at 2:50 | comment | added | Richard Diagram | @PaulBlainLevy I agree, but I'm not the best at this sort of stuff, lol. | |
| Aug 18, 2021 at 2:45 | comment | added | Paul Blain Levy | @RichardDiagram this looks $\Pi^1_1$ to me, but I can't say for certain. | |
| Aug 18, 2021 at 1:45 | comment | added | Richard Diagram | I'm curious as to how this works exactly. For all analytic functions $F(z) : \mathcal{J} \to \mathbb{C}$ such that $F(e^z) = F(z)+1$ and $F(0) = -1$, then $F$ must have the form $\text{slog}(z)$. Where here $\text{slog}$ is Kneser's slog and $\mathcal{J}$ is the julia set of $\exp$ excluding periodic points. This is an open problem in iteration theory. I'm wondering if this is along the lines. There is only one quantifier as I understand it, but I'm not too sure. | |
| Aug 17, 2021 at 20:41 | comment | added | Paul Blain Levy | @NoahSchweber done. Thanks for the suggestion! | |
| Aug 17, 2021 at 20:35 | history | edited | Paul Blain Levy | CC BY-SA 4.0 |
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| Aug 17, 2021 at 19:48 | comment | added | Noah Schweber | A quick suggestion: to avoid clutter, maybe the candidates which turned out to have arithmetical equivalents should be deleted? | |
| Aug 17, 2021 at 16:20 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Aug 17, 2021 at 13:47 | vote | accept | Paul Blain Levy | ||
| Aug 17, 2021 at 13:25 | history | edited | Denis Serre | CC BY-SA 4.0 |
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| Aug 17, 2021 at 13:22 | answer | added | Paul Blain Levy | timeline score: 4 | |
| Aug 17, 2021 at 11:55 | history | edited | Paul Blain Levy | CC BY-SA 4.0 |
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| Aug 17, 2021 at 11:40 | history | edited | Paul Blain Levy |
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| Aug 16, 2021 at 20:14 | history | edited | Paul Blain Levy | CC BY-SA 4.0 |
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| Aug 16, 2021 at 20:09 | history | edited | Paul Blain Levy | CC BY-SA 4.0 |
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| Aug 15, 2021 at 4:41 | comment | added | user44143 | @Wojowu, I gave an arithmetic translation of Schanuel’s conjecture in my answer. | |
| Aug 14, 2021 at 23:25 | vote | accept | Paul Blain Levy | ||
| Aug 16, 2021 at 19:44 | |||||
| Aug 12, 2021 at 17:18 | comment | added | Wojowu | @MattF. That's a fair point. Can you think of a way to phrase, say, Schanuel's conjecture in arithmetic way? | |
| Aug 11, 2021 at 15:10 | comment | added | Terry Tao | There are (open) special cases of the invariant subspace conjecture that are of the form requested by the OP, e.g., Conjecture 8 of terrytao.wordpress.com/2010/06/29/… . | |
| Aug 11, 2021 at 10:11 | comment | added | Paul Blain Levy | @TimCampion a continuous function can be presented as a continuous function on rationals. | |
| Aug 9, 2021 at 15:52 | comment | added | Tim Campion | Sorry, what is meant by "genuine" quantifier? Doesn't the term "continuous function" implicitly contain quite a few quantifiers? (EDIT: Oh I see -- the condition that a given function be continuous is arithmetical, so the only analytic quantifier is over the function itself) | |
| Aug 9, 2021 at 15:35 | comment | added | Wojowu | I suspect the four exponentials conjecture and some other problems in transcendental number theory could qualify. | |
| Aug 8, 2021 at 21:21 | comment | added | none | Oops, yes, I can't edit the comment any more though. This thread's answers have some more possibilities. Is there a theorem of Takeuti that practically everything in classical analysis can be encoded in Peano arithmetic? That might make this question difficult. | |
| Aug 8, 2021 at 19:14 | comment | added | Joel David Hamkins | @none I think you mean to say that RH has a $\Pi^0_1$ form, that is, a purely arithmetic form as a universal arithmetic assertion. | |
| Aug 8, 2021 at 17:02 | comment | added | none | $\forall (x,y).\zeta(x+iy)=0\implies x={1\over 2}$ is a natural thing to write, but RH turns out to have a $\Pi^1_0$ form. So your question has subtleties. | |
| Aug 8, 2021 at 12:18 | comment | added | Joel David Hamkins | Problems of the form: is there a countable graph with such-and-such first-order property? Is there a countable structure with such-and-such first-order property? Negative answers would have the desired form $\Pi^1_1$. | |
| Aug 8, 2021 at 12:06 | comment | added | Gerry Myerson | I don't know, but you could have a look at tandfonline.com/doi/abs/10.1080/… and math.ksu.edu/~ramm/papers/547.pdf and math.stackexchange.com/questions/1095743/… and math.stackexchange.com/questions/58638/… and link.springer.com/article/10.1007/s00020-018-2460-8 and mathoverflow.net/questions/100265/… | |
| Aug 8, 2021 at 10:43 | history | asked | Paul Blain Levy | CC BY-SA 4.0 |