Timeline for Euler's constant: irrationality and proof theory
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2015 at 19:46 | comment | added | Christian Remling | @Wojowu: This issue is also discussed in the Wikipedia article. Chaitin's $\Omega$ is also non-computable in the "modern sense" (Wikipedia) because it's irrational, so good enough rational approximations will eventually determine its digits. | |
| Nov 29, 2015 at 19:44 | comment | added | Wojowu | @ChristianRemling Ah, that changes everything. I thought that definition of computable number is a number having computable string as binary expansion (which, as I argue, is not obvious to prove). Sorry for all the confusion! | |
| Nov 29, 2015 at 19:41 | comment | added | Christian Remling | @Wojowu: en.wikipedia.org/wiki/Computable_number#Formal_definition | |
| Nov 29, 2015 at 19:40 | comment | added | Wojowu | @ChristianRemling This isn't enough. For example, if we look at $1$ instead of $\gamma$ and we have (in binary) $a_n=0.11...1$ ($n+1$ ones) then $|a_n-1|<2^{-n}$, but digits of $a_i$ don't match up with digits of $1=1.000...$. | |
| Nov 29, 2015 at 19:37 | comment | added | Christian Remling | @Wojowu: $|a_n-\gamma|<2^{-n}$ | |
| Nov 29, 2015 at 19:35 | comment | added | Wojowu | @ChristianRemling What do you mean by "controlled errors"? | |
| Nov 29, 2015 at 19:34 | comment | added | Christian Remling | @Wojowu: $\gamma$ is in fact computable (and this is what we need here); this means that a Turing machine can compute rational approximations to $\gamma$ with controlled errors, which is obviously the case here. | |
| Nov 29, 2015 at 18:29 | comment | added | Wojowu | I was thinking of this example, but I didn't post this, because I actually wasn't sure if $\gamma$ is uncomputable. For what I know, $\gamma$ might have some incredibly close dyadic fraction approximations, and they might be so close that to determine whether part of expansion is $01111...$ or $10000...$ might require uncomputable degree of accuracy. Although by no means I expect this to be the case, I don't see how such possibility could be excluded without some bounds on the approximation of $\gamma$ by rationals, and afaik none are known. | |
| Nov 29, 2015 at 15:19 | history | answered | Adam P. Goucher | CC BY-SA 3.0 |