## Some fundamental issues concerning degrees of unsolvability (2007)

Venue: | In [6], 2005. Preprint |

Citations: | 9 - 8 self |

### BibTeX

@INPROCEEDINGS{Simpson07somefundamental,

author = {Stephen G. Simpson},

title = {Some fundamental issues concerning degrees of unsolvability},

booktitle = {In [6], 2005. Preprint},

year = {2007},

pages = {53--462}

}

### OpenURL

### Abstract

Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.