Minimality Arguments for Infinite Time Turing Degrees
PHILIP D. WELCH · Cambridge University Press eBooks · 1999
We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1-degrees of certain \\Sigma 1 2 -correct KP-models of the form L oe (oe ! ! L 1 ) have minimal upper bounds. 1 Introduction Obtaining minimality results in degree theory has a long history: the methods go back to those of Spector when he constructed minimal Turing degrees, and to Gandy-Sacks, [2], for minimal hyperdegrees. The perfect set construction is the common thread to these proofs. A further feature, which is shared, either directly or indirectly, by such arguments, is the use of a selection principle in order to typically, directly shrink a perfect set T ` ! ! to a T 0 so that a particular function is either continuous one-to-one, or constant on the branches of T 0 . For ex...