Timeline for Martin's cone theorem and recursion theory
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 7, 2011 at 5:23 | vote | accept | Andrés E. Caicedo | ||
| Apr 13, 2011 at 9:07 | answer | added | 喻 良 | timeline score: 7 | |
| Nov 19, 2010 at 5:33 | comment | added | Andrés E. Caicedo | @Bjørn, you are right, the Reimann-Slaman results do not give examples. | |
| Nov 18, 2010 at 20:24 | comment | added | Bjørn Kjos-Hanssen | If we just want a set that contains no largest cone (i.e., no smallest base) then we can take the nonzero degrees, i.e., those $\mathbf a$ with $\mathbf a>\mathbf 0$. | |
| Nov 18, 2010 at 18:31 | comment | added | Bjørn Kjos-Hanssen | Reimann and Slaman show that there is a cone consisting of continuously random reals. However they then show that every non-$\Delta^1_1$ real is continuously random. So Kleene's $\mathcal O$ is a fairly natural base for such a cone. | |
| Nov 18, 2010 at 11:17 | answer | added | Bjørn Kjos-Hanssen | timeline score: 4 | |
| Nov 10, 2010 at 19:05 | comment | added | Andrés E. Caicedo | @Henry: Thanks! That's a good place to start. | |
| Nov 10, 2010 at 19:04 | history | edited | Andrés E. Caicedo | CC BY-SA 2.5 |
added 9 characters in body
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| Nov 10, 2010 at 18:31 | comment | added | Henry Towsner | Slaman and Reimann have some results on random reals (arxiv.org/abs/0707.1390) which use the cone theorem, where it looks like they have some information about the cones, but not an exact base. | |
| Nov 10, 2010 at 17:43 | comment | added | Joel David Hamkins | I heard Martin say at a talk once that when he proved his theorem, it was easy to think at first that perhaps it might be used to refute AD, since after all, he knew lots of sets of Turing degrees, and all that was required was to find one such set that neither contained nor omitted a cone. But of course, it didn't turn out that way... | |
| Nov 10, 2010 at 17:22 | history | asked | Andrés E. Caicedo | CC BY-SA 2.5 |