The Strongest Nonsplitting Theorem

Sacks [14] showed that every computably enumerable (c.e.) degree ≥ 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington [8] showed that one could take a = 0′. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0′. Heterogeneous splittings are best considered in the context of cupping and noncupping. Posner and Robinson [13] showed that every nonzero Δ2 degree can be nontrivially cupped to 0′, and Arslanov [1] showed that every c.e. degree > 0 can be d.c.e. cupped to 0′ (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree > 0 is d.c.e. cuppable.) Cooper [2] and Yates (see Miller [11]) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper [3], and Slaman and Steel [15] (see also Downey [7]), who showed that there is a nonzero c.e. degree a below which even Δ2 cupping of c.e. degrees fails. We prove below what appears to be the strongest possible of such nonsplitting and noncupping results. Theorem 1. There exists a computably enumerable degree a < 0′ such that there exists no nontrivial cuppings of c.e. degrees in the Δ2 degrees above a.