Timeline for Existence of a set of valid Busy-Beaver entries.
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Jun 29, 2011 at 12:43 | answer | added | sligocki | timeline score: 2 | |
| Jun 23, 2011 at 16:56 | vote | accept | Saran Neti | ||
| Jun 23, 2011 at 16:22 | vote | accept | Saran Neti | ||
| Jun 23, 2011 at 16:56 | |||||
| Jun 23, 2011 at 16:22 | comment | added | Saran Neti | @mhum The Wikipedia article on Intuitionism certainly helps. Finitism rejects the existence of countably infinite, Intuitionism rejects the existence of uncountably infinite; I was looking for a constructive theory that accepts the uncountably infinite but rejects the non-computable. But, thanks! | |
| Jun 23, 2011 at 11:47 | comment | added | Todd Trimble | @mhum: that wikipedia article looked like it had problems. One can definitely prove that the power set of the naturals is uncountable in intuitionistic logic. (Notice that "X is uncountable" is a negated sentence, a proposition of the form $q = \neg p$. For such $q$, one has $q$ equivalent to $\neg \neg q$.) | |
| Jun 23, 2011 at 4:16 | comment | added | mhum | @sarannmr: If you do not accept the existence of the non-computable, then you must also reject the existence of uncountable things. Without the uncountable, you do not have the real numbers, only the rationals. This rejection is apparently also a feature of intuitionism (en.wikipedia.org/wiki/Intuitionism). Maybe worth a look? | |
| Jun 23, 2011 at 3:11 | comment | added | Saran Neti | Gerhard : What bothers me is that I cannot understand the meaning of existence outside of computability. If you could point to an actual weak axiomatic system with its limited scope, I'd be very interested to know some simple statements that cannot be expressed in them, but intuitively, I should be able to. i.e what prompts the bootstrapping of properties that render axiomatic systems victims of Godel's incompleteness theorems. | |
| Jun 23, 2011 at 1:20 | answer | added | Joel David Hamkins | timeline score: 11 | |
| Jun 23, 2011 at 0:53 | history | edited | Saran Neti | CC BY-SA 3.0 |
edited body
|
| Jun 23, 2011 at 0:42 | comment | added | Gerhard Paseman | You can say (to yourself) whatever pleases you. For Question 1, s exists iff M halts when given a blank tape as input. There are machines that halt on 1 step which have n states, so E_n is nonempty for n large enough (n >0?). That you can't determine its members in your lifetime may be troubling, but so what? There are axiomatic systems where everything that exists are computable, but they are limited in scope. You might ponder more about what is bothering you, and see if you can turn that into a question. Gerhard "Believes No Impossibilities Before Coffee" Paseman, 2011.06.22 | |
| Jun 23, 2011 at 0:23 | history | asked | Saran Neti | CC BY-SA 3.0 |