Timeline for How does the Constructibility Degree of a real compare with its Turing Degree?
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| Oct 24, 2010 at 16:42 | comment | added | Andrés E. Caicedo | @Tanmay, you may also want to contact directly Marcia Groszek or François Dorais for more up-to-date results, and perhaps links to surveys or introductory material. | |
| Oct 24, 2010 at 16:38 | comment | added | Andrés E. Caicedo | @Tanmay, keep in mind that most results in the constructibility side of this area require forcing. They may establish the corresponding result for Turing degrees, or there may be subtle differences, and in that case, the recursion theoretic counterpart tends to require a priority argument. In any case, there are some basic results that can be read without too much background. A good starting point is a survey by Richard Shore, "Degrees of constructibility", in Set theory of the continuum (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., 26, Springer, New York, 1992, 123–135. | |
| Oct 21, 2010 at 19:16 | comment | added | user3462 | Hi Stefan, thanks for the answer. This answers the more general question which I unfortunately only mentioned in the title. Andres, if you know of any suitably low-level(in the sense of starting from a modest background/survey article) paper of theirs, could you please send me a link? | |
| Oct 20, 2010 at 21:39 | comment | added | Andrés E. Caicedo | Anyway, even if $\aleph_1^{L[a]}$ is countable for all $a$, constructibility degrees are much coarser than Turing degrees. Note $a$ is constructibly equivalent to its jump, its Hyperjump, and much more. Grozsek and her co-authors and students have looked at the relation between the order of constructible degrees and the Turing degrees (they are not equivalent structures). | |
| Oct 20, 2010 at 21:32 | history | answered | Stefan Geschke | CC BY-SA 2.5 |