DEGREE INVARIANCE IN THE Pi(0)(1) CLASSES

作者：Weber Rebecca^*

来源：Journal of Symbolic Logic, 2011, 76(4): 1184-1210.

Let l Pi denote the collection of all Pi(0)(1) classes, ordered by inclusion. A collection of Turing degrees l is called invariant over l Pi if there is some collection S of Pi (0)(1)classes representing exactly the degrees in l such that S is invariant under automorphisms of l Pi. Herein we expand the known degree invariant classes of l Pi, previously including only {0} and the array noncomputable degrees, to include all high,, and non-low degrees for n >= 2. This is a corollary to a very general definability result. The result is carried out in a substructure G of l Pi, within which the techniques used model those used by Cholak and Harrington [6] to obtain the same definability for the c.e. sets. We work back and forth between G and l Pi to show that this definability in G gives the desired degree invariance over l Pi

• 出版日期2011-12