Uniformly defined descending sequences of degrees

Abstract

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π11 sets of natural numbers are precisely those which are defined by a Σ11 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each xn+ 1 is a set in the ω-model coded by xn, and each xn+1 is the unique solution of P(xn, xn+1) for some fixed arithmetic P.