
doi: 10.2307/2273788
In the December 1982 issue of this Journal Weitkamp [W] posed some questions concerning the incomparability of certain “r.e.” sets for the notion of Kleene reducibility. He asked whether the incomparability of, for example, the Friedman set F (defined below) and the set WI0 (the set of reals coding wellfounded trees of admissible height) was equivalent to the existence of 0#, since forcing over L with a set of conditions could not achieve this. We answer this by showing that in a certain class generic extension of L they are comparable, but 0# does not exist. This is an application of Jensen's coding theorem (cf. [BJW]), using a modified construction due to René David [D]. Indeed the result here is a simple application of his result. Define F as follows:Harrington showed, in effect, that one could not add a cone of Turing degrees to this set by forcing with sets of conditions over L. The method used here does add a cone of Turing degrees to a much simpler set RI1 (defined below)—and indeed the whole process could be viewed as forcing over L to obtain the determinacy of certain rather simple sets. It is the determinacy of the game with payoff set RI1 that ensures the comparability of F and WI0 (amongst many others).We shall refrain from repeating all the basic definitions and lemmas since the reader can readily refer to [W]; we shall give the basic necessities.
Determinacy principles, Kleene reducibility, class forcing, incomparability, sharps, set of reals coding well founded trees of admissible height, Other set-theoretic hypotheses and axioms, Higher-type and set recursion theory, Friedman set, Other degrees and reducibilities in computability and recursion theory, Consistency and independence results, determinacy, Descriptive set theory
Determinacy principles, Kleene reducibility, class forcing, incomparability, sharps, set of reals coding well founded trees of admissible height, Other set-theoretic hypotheses and axioms, Higher-type and set recursion theory, Friedman set, Other degrees and reducibilities in computability and recursion theory, Consistency and independence results, determinacy, Descriptive set theory
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