Timeline for Simple example of a sequence without computable modulus of convergence
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2011 at 5:09 | answer | added | Jason | timeline score: 1 | |
| Jan 12, 2011 at 13:29 | vote | accept | Mateus Araújo | ||
| Jan 12, 2011 at 1:08 | comment | added | Zen Harper | Yes, not my construction of course; E. Bishop, Brouwer and the constructivists have been doing that kind of stuff for a long time. As they pointed out, if anyone ever does solve the Goldbach Conjecture, you can just immediately switch to any other similar open problem; so these things will always be around. | |
| Jan 12, 2011 at 1:02 | history | edited | Mateus Araújo | CC BY-SA 2.5 |
elaborating on question
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| Jan 12, 2011 at 1:00 | comment | added | Mateus Araújo | That's an interesting construction, in that many open problems in mathematics can be mapped to the convergence of a sequence, in a way that does not help at all in solving the original problem. | |
| Jan 12, 2011 at 0:35 | comment | added | Zen Harper | ...(continued) Most mathematicians believe that the limit and $N$ are a computable number and function, but we just don't know which; constructivists would argue that the limit doesn't even exist. So, in both cases (although maybe constructivists wouldn't allow excluded middle here?!!!!), this isn't really an answer to your precise question, sorry. But I thought it might be interesting. | |
| Jan 12, 2011 at 0:34 | comment | added | Zen Harper | Not really an answer, so only a comment (DISCLAIMER: I know almost nothing about logic and computability theory), but how about the following: $a_n = 0$ if all even integers $k$ with $4 \leq k \leq n+4$ can be written as the sum of two primes, and $a_n = 1$ otherwise. Then $(a_n)$ is increasing, and converges to either $0$ or $1$ (classically, anyway). Unfortunately, we'll never know which until someone solves the Goldbach Conjecture; and moreover, even knowing which one it is might not be enough to calculate $N(\epsilon)$ explicitly...(to be continued...) | |
| Jan 11, 2011 at 23:43 | answer | added | Andreas Blass | timeline score: 9 | |
| Jan 11, 2011 at 23:28 | history | asked | Mateus Araújo | CC BY-SA 2.5 |