The density of the low2 n-r.e. degrees

Abstract

Although density fails in the d-recursively enumerable (d-r.e.) degrees, and more generally in the n-r.e. degrees (see [3]), we show below that the low 2 n-r.e. degrees are dense. This is achieved by combining density and splitting in the manner of Harrington’s proof (see [9]) for the r.e. (=1r.e.) case. As usual we use low 2 -ness to eliminate infinite outcomes in the construction, with the result that the proof is simpler than that of the general splitting theorem [2] for the n-r.e. degrees, which unlike the Sacks splitting theorem ([7] or p. 124 of [11]) was infinite injury. By working below a low 2 degree we avoid the original device in such theorems (see Robinson [6] or p. 224 of [11]) dependent on the recursion theorem. (We note however that Watson [12] proved a suitable version of the fixed-point theorem for d-r.e. sets which could be used for such a proof for the low2 d-r.e. degrees). For other recent results concerning joins and density in the d-r.e. and n-r.e. degrees see [1], [4] and [5].