Vol. 18, No. 2, 1966

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Many-one degrees of the predicates Ha (x)

Yiannis (John) Nicolas Moschovakis

Vol. 18 (1966), No. 2, 329–342

Spector proved in his Ph. D. Thesis that if |a| = |b|(a,b O), then Ha(x) and Hb(x) have the same degree of unsolvability; Davis had already shown that if |a| = |b| < ω2, then Ha(x) and Hb(x) are in fact recursively isomorphic, i.e.,

Ha(x) ≡ Hb (f (x)),

where f(x) is a recursive permutation.

In this note we prove that if |a| = |b| = ξ, then Ha(x) and Hb(x) need not have the same many-one degree, unless ξ = 0 or is of the form η + 1 or η + ω; if ξ0 is not of the form η + 1 or η + ω, then the partial ordering of the many-one degrees of the predicates Ha(x) with |a| = ξ contains well-ordered chains of length ω1 as well as incomparable elements. The proof rests on a combinatorial result which relates the many-one degree of Ha(x)(a= 3.5a O) to the rate with which the sequence of ordinals |an| approaches |a′|.

Mathematical Subject Classification
Primary: 02.77
Received: 26 December 1964
Published: 1 August 1966
Yiannis (John) Nicolas Moschovakis