Computable trees, prime models, and relative decidability

Abstract

Abstract. We show that for every computable tree T with no dead ends and all paths computable, and every D > T ∅, there is a D-computable listing of the isolated paths of T . It follows that for every complete decidable theory T such that all the types of T are computable and every D > T ∅, there is a D-decidable prime model of T . This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.

The atomic model theorem and type omitting

“…Improving a result of Csima [8], Hirschfeldt [20] showed that Theorem 5.2 cannot be improved, in the following sense. However, outside the ∆ 2 degrees, the situation for type omitting is different from that for atomic models.…”

“…Furthermore, each path of T (other than 0 ω ) is a type of T equivalent to one of the rows of S, and hence is recursive in S. So for each path P of T we have D T P . It is easy to check that the proof of Theorem 2.1 of Hirschfeldt [20] can now be carried out in RCA 0 to show that there is a D-recursive listing of the isolated paths of T , and hence a D-recursive listing of the principal types of T . It follows from Theorem 6.2 that T has an atomic model.…”

Relative to Any Non-Hyperarithmetic Set

“…Slaman [Sla98] and Wehner [Weh98] independently proved the following theorem (with a third proof later provided by Hirschfeldt [Hir06]).…”

A characterization of the 0-basis homogeneous bounding degrees

“…Many people including Csima, Hirschfeldt, Knight, Soare, and Epstein have recently studied the prime case in [3], [10], [5], and [6]. In [5], Csima, Hirschfeldt, Knight, and Soare characterized the prime bounding degrees.…”