Timeline for How undecidable is the spectral gap?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Dec 14, 2015 at 23:23 | history | edited | Conifold | CC BY-SA 3.0 | added 680 characters in body |
| Dec 13, 2015 at 23:20 | vote | accept | Conifold | ||
| Dec 13, 2015 at 14:05 | comment | added | Steven Gubkin | Scott Aaronson has some nice commentary here, including my worry about "hypercomputing": scottaaronson.com/blog/?p=2586 | |
| Dec 13, 2015 at 13:59 | comment | added | Steven Gubkin | @QiaochuYuan Ah I see. I thought that, perhaps, after setting up a given system measuring the spectral gap would be a physical property of the system which could be measured instantly. | |
| Dec 13, 2015 at 3:22 | comment | added | Qiaochu Yuan | So there's no hope for learning about the truth of CH this way, only its provability with respect to various sets of axioms. At best the statements you can attempt to learn the truth about are statements of the form "this Turing machine eventually halts," and you won't be able to put a computable bound on how long you need to wait to learn the truth of these statements. You would also be much better off running these Turing machines on your computer than via a spectral gap problem. | |
| Dec 13, 2015 at 3:15 | comment | added | Qiaochu Yuan | @Steven: no, that can't happen. As far as I can tell, the way you encode that in a spectral gap problem is by using the spectral gap problem to simulate a Turing machine searching for a proof of CH in ZFC. Since we know that CH is independent of ZFC, we already know what will happen: the Turing machine will run forever and won't halt, and we won't have learned anything from this (except, if we really are willing to wait literally forever, that ZFC is consistent). | |
| Dec 13, 2015 at 2:57 | comment | added | Steven Gubkin | @QiaochuYuan I am not quite sure. Maybe some undecidable sentence like "Continuum hypothesis is true" can be coded into this lattice, and then you can instantly read off the "truth" of CH by looking for a spectral gap. This sounds pretty silly, but this kind of interplay is interesting to me. | |
| Dec 13, 2015 at 1:56 | comment | added | Fan Zheng | @QiaochuYuan If that program really terminates, is it more likely because of an inconsistency in ZFC or a bug in the program? | |
| Dec 12, 2015 at 13:00 | answer | added | Joel David Hamkins | timeline score: 20 | |
| Dec 12, 2015 at 8:47 | answer | added | Qiaochu Yuan | timeline score: 8 | |
| Dec 12, 2015 at 8:41 | comment | added | Qiaochu Yuan | @Steven: it's the same thing that would happen if you wrote a program to search for a proof of a contradiction in ZFC... | |
| Dec 12, 2015 at 2:49 | answer | added | Henry Towsner | timeline score: 38 | |
| Dec 11, 2015 at 22:34 | answer | added | Stopple | timeline score: 5 | |
| Dec 11, 2015 at 22:15 | comment | added | Steven Gubkin | I also wonder, if they prove that a particular setup is undecidable, what happens when they actually physically construct that setup and measure the spectral gap? Do they gain truth about mathematics inaccessible to logic? | |
| Dec 11, 2015 at 22:08 | history | asked | Conifold | CC BY-SA 3.0 |