Recursive ordinal

Abstract: This paper completes an investigation of «jumps» of orderings. The last few cases are given in the proof that for each recursive ordinal α≥1 and for each Turing degree d≥0 (α) , there is a linear ordering A such that d is least among the αth jumps of degrees of (open diagrams of) isomorphic copies of A, and for β<α, the set of βth jumps of degrees of copies of A has no least element

Abstract: In [3] Friedberg showed that every Turing degree 0' is the jump of some degree. Using the relativized version of this theorem it can be shown by finite induction that if a >-8 then there is a b such that bn) = a (our notation is defined in ?1). It is natural to ask whether these results can be extended into the transfinite. Is it true for example, that whenever a 0(w) there is a b such that b ') = a? In ?2 we use forcing to prove this result. (The usefulness of forcing in answering questions involving the jump operation on degrees of unsolvability has been previously demonstrated in Selman [5] where, for example, forcing was used to construct degrees a and b such that a') = bw) = a U b = 0(-).) In ?3 we generalize the methods of ?2 to show that if a is a recursive ordinal and a 0(a) then there is a b such that b'a) = a, i.e. the Friedberg result can be extended to all recursive ordinal levels. Thomason [6] used a forcing argument to show: If A ?hey (the Kleene set of notations for the recursive ordinals) then there is a B such that A-h6B (the set of notations for ordinals recursive in B). In ?4 we show this result holds when hyperarithmetic reducibility is replaced by Turing reducibility: If A ?TC then there is a B such that A =TJB.

Abstract: Solovay has shown that if F: [)]' -2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every o0, where a is a recursive ordinal, there is a clopen partition F: [)]' -2 such that every infinite homogeneous set is Turing above o0 (an anti-basis result). Here we refine these results, by associating the "order type" of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem. In [20] R. Solovay analyzed the effective version of the Galvin-Prikry generalization of Ramsey's theorem w Borel (w))'. Among other results, Solovay proved the basis theorem for the clopen Ramsey theorem: if F: [w)]0 -+ 2 is a recursive partition (i.e. clopen with recursive code) then there is an infinite hyperarithmetic homogeneous set X c w). In [17] S. G. Simpson proved an antibasis theorem for the clopen Ramsey theorem: given any recursive ordinal a there is a recursive partition F: [wo]'o 2 such that O' is recursive in every infinite homogeneous set X for F. This was an important result which led to the discovery in [8] of a combinatorial first-order Paris-Kirby-type statement which is independent of the theory ATRo (a subsystem of second-order arithmetic which is weaker than 111comprehension yet much stronger than first-order Peano arithmetic). In this paper we present a finer recursion theoretic analysis of the clopen Ramsey theorem, correlating the "order type" of a clopen partition and the Turing degrees of the infinite sets homogeneous for the partition. In [4] (see [3], which was written without knowledge of Simpson's results) we established anti-basis results for the Nash-Williams barrier theorem, a generalization of Ramsey's theorem which is very close to the clopen Ramsey theorem. The techniques developed for the barrier theorem adapt immediately to yield exactly the same results for the clopen Ramsey Received June 20, 1982. 1980 Mathematics Subject Classification: 03D25, 03D30.