Vol. 31, No. 1, 1969

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Cohesive sets and recursively enumerable Dedekind cuts

Robert Irving Soare

Vol. 31 (1969), No. 1, 215–231

In this paper the methods of recursive function theory are applied to certain classes of real numbers as determined by their Dedekind cuts or by their binary expansions. Instead of considering recursive real numbers as in constructive analysis, we examine real numbers whose lower Dedekind cut is a recursively enumerable (r.e.) set of rationals, since the r.e. sets constitute the most elementary nontrivial class which includes nonrecursive sets. The principal result is that the sets A of natural numbers which “determine” such real numbers α (in the sense that the characteristic function of A corresponds to the binary expansion of α) may be very far from being r.e., and may even be cohesive. This contrasts to the case of recursive real numbers, where A is recursive if and only if the corresponding lower Dedekind cut is recursive.

Mathematical Subject Classification
Primary: 02.70
Received: 27 June 1967
Revised: 18 February 1969
Published: 1 October 1969
Robert Irving Soare