The Degree of Unsolvability of a Real Number

Abstract

Each real number x can be assigned a degree of unsolvability by using, for example, the degree of unsolvability of its binary or decimal expansion, or of its Dedekind cut, or of some other representation of x. We show that the degree of unsolvability assigned to x in any such way is the same regardless of the representation used. This gives to each real number a unique degree of unsolvability. If x is of computably enumerable degree, there is a computable sequence of rationals which converges to x with a modulus of convergence having the same degree of unsolvability as x itself. In contrast, if x is computable relative to the halting set but is not of computably enumerable degree, this is not true. Specifically, if r _n is any computable sequence of rationals converging to such a real number x, the modulus of convergence of r _n must have degree of unsolvability strictly higher than that of x. Thus there is an inherent gap between the degree of unsolvability of such an x and the degree of unsolvability of the modulus of convergence of an approximating computable sequence of rationals; this gap is bridged (in the sense of the “join operator” of degree theory) by a set of natural numbers which measures the twists and turns of the computable sequence r _n.

Please address all correspondence to M. B. Pour-El. The authors are grateful to the referees for simplifications of two of the proofs.