Non-cuppable enumeration degrees via finite injury

Abstract

We exhibit finite injury constructions of a high Σ 0 2 enumeration degree incomparable with all intermediate ∆ 0 2 enumeration degrees, as also of both an upwards properly Σ 0 2 high and a low 2 noncuppable Σ 0 2 enumeration degree †. We also outline how to apply the same methods to prove that, for every Σ 0 2 enumeration degree b there exists a noncuppable degree a such that b ≤ a and a ≤ b , thus showing that there exist noncuppable Σ 0 2 enumeration degrees at every possible level of the high/low jump hierarchy.

Badness and jump inversion in the enumeration degrees

“…Griffith's technique for inverting the jump in the enumeration degrees lends itself easily to certain constructions in the 0 2 degrees as shown, for example, in [9,13] and [12]. For example in [12] the present author shows, using a finite injury oracle construction (with K as oracle), the existence of a high degree a < 0 e such that a is incomparable with any intermediate (i.e.…”

“…For example in [12] the present author shows, using a finite injury oracle construction (with K as oracle), the existence of a high degree a < 0 e such that a is incomparable with any intermediate (i.e. nonzero and incomplete) 0 2 degree.…”

“…The proof used in Sect. 3 is a finite injury priority argument in oracle K (the Turing halting set) and applies in the context of the 0 2 enumeration degrees similar techniques to those pioneered by the author in the 0 2 enumeration degrees [12,13]. The reader may wish to compare this to the infinite injury priority argument, also using oracle K, used by Calhoun and Slaman [6] in the proof of nondensity in the 0 2 degrees mentioned above.…”

“…This made available tools used later in [13] and [12] for constructing enumeration degrees with specific relative jump complexities. For example, Theorem 2.3 in [13] shows that, given any enumeration degree b ≤ e 0 e there exists a noncuppable degree 0 e < a < 0 e -i.e.…”