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Local Density of Kleene Degrees

✍ Scribed by Hisato Muraki

Publisher
John Wiley and Sons
Year
1995
Tongue
English
Weight
302 KB
Volume
41
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

Concerning Post's problem for Kleene degrees and its relativization, Hrbacek showed in [1] and [2] that if V = L, then Kleene degrees of coanalytic sets are dense, and then for all K βŠ†^Ο‰^Ο‰, there are N~1~ sets which are Kleene semirecursive in K and not Kleene recursive in each other and K. But the density of Kleene semirecursive in K Kleene degrees is not obtained from these theorems. In this note, we extend these theorems by showing that if V = L, then for all K βŠ† ^Ο‰^Ο‰, Kleene semirecursive in K Kleene degrees are dense above K.

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## Abstract In this note, we study the complementedness and the distributivity of upper semilattices of Kleene degrees assuming __V = L. K__ denotes the upper semilattice of all Kleene degrees. We prove that if __V = L__, then some sub upper semilattices of __K__ are non‐complemented and some are n