Timeline for Approaches to Riemann hypothesis using methods outside number theory
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Aug 7, 2010 at 8:43 | comment | added | KConrad | An unprovable statement about a rapidly growing function (like the Ackerman function) which has no natural meaning in number theory does not satisfy me as being comparable to RH. | |
| Aug 7, 2010 at 8:37 | comment | added | KConrad | That survey has no statements I would consider number-theoretically interesting. The section on pp. 7--8 has some statements about the zeta-function, but none are naturally interesting on their own. The reference to Schinzel's Hypothesis H possibly being unprovable in PA also strikes me as completely speculative (along the lines of "we can't prove it, so maybe it's unprovable", with no more substantive rationale). I know Friedman has been trying for many years to find naturally interesting statements in number theory that are of broad interest and indep. of PA. AFAIK, the search continues. | |
| Aug 7, 2010 at 6:17 | comment | added | Junkie | Some "unprovable in PA" statements seem to relate to some function growing too fast, even if not explicitly. $$ $$ Here is a 2006 survey on "Brief introduction to unprovability" cs.umd.edu/~gasarch/largeramsey/bovINTRO.pdf | |
| Aug 7, 2010 at 6:11 | comment | added | Junkie | Proving that things are unprovable in PA was already a hassle. See section 3 of mathdl.maa.org/images/upload_library/22/Ford/Spencer669-675.pdf $$ $$ "The statements constructed by Godel suffered the defect of being unnatural and for the past 50 years a somewhat raggedy debate ensured concerning whether or not his result applied to statements of real mathematical interest. In 1977, Paris/Harrington gave the first "natural" example of a statement that was true for $Z$ but unprovable in PA, [coming from Ramsey theory]." I do not know the current status of "not provable in PA" statements. | |
| Aug 7, 2010 at 5:45 | comment | added | BCnrd | OK, now that the except from Shelah's writing is given, I see there's no underlying idea at all; it is a speculative dream, so to speak of it being "promising" to yield insight on RH is putting the cart before the horse. In view of the fact that people have proved RH-type results in other settings (via the hard and brilliant work of Grothendieck, Artin, Deligne, etc.), I see no reason to take this dream seriously. It reminds me of the speculations "maybe FLT is unprovable?" before 1993, which were all forgotten after Wiles contributed real ideas to solve the problem. | |
| Aug 7, 2010 at 5:39 | comment | added | KConrad | Has anything remotely like RH (on locations of zeros of a non-artificial function) been shown to be unprovable in PA? This dream sounds to me like the old canard that when you have a hammer everything looks like a nail. | |
| Aug 7, 2010 at 2:50 | history | edited | Junkie | CC BY-SA 2.5 |
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| Aug 7, 2010 at 0:11 | comment | added | BCnrd | Is there actually an idea here, or just a dream? | |
| Aug 6, 2010 at 22:17 | comment | added | Joseph O'Rourke | I would be grateful for any (one-two sentence) summary of Shelah's idea, which is not easy (for me) to extract from his Logical Dreams. | |
| Aug 6, 2010 at 17:59 | history | edited | doetoe | CC BY-SA 2.5 |
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| Aug 6, 2010 at 16:09 | history | answered | doetoe | CC BY-SA 2.5 |