Sacks forcing sometimes needs help to produce a minimal upper bound

Extract

Does every countable set of hyperdegrees have a minimal upper bound?

This question remains unanswered. In this paper, we extend the known results.

The standard way to construct minimal upper bounds for degrees is to force with pointed perfect trees. This works for hyperdegrees, in the right context. Sacks [Sa] showed that if an admissible set A satisfies Σ₁ DC, then forcing with its uniformly hyperarithmetically pointed perfect trees yields a minimal upper bound for the degrees in A.

A next question is whether Σ₁DC is necessary. Abramson [A] built an admissible set such that Sacks forcing, or anything like it, would not produce a minimal upper bound. He left open the question, though, whether there is such a bound for his set.

We answer this question affirmatively.

In §II we summarize the previous relevant results, including Steel forcing. In §III we give a construction different from Abramson's of an admissible set for which Sacks forcing does not produce the desired bound. We present this alternative because it is different (although still based on Steel forcing), simpler than the original, and fully illustrates the technique of finding the bound, which applies equally well to the earlier example. §IV describes the construction of the bound. §V closes with questions.

I thank Professor Sy Friedman for bringing this problem to my attention. This paper is dedicated to Professor Alexander Kechris on the occasion of his fortieth birthday, and to the Los Angeles VIGOL on its tenth anniversary.