The Decidability of the Existential Theory of the Poset of Recursively Enumerable Degrees with Jump Relations

Abstract

We show that the existential theory of the recursively enumerable degrees in the language L containing predicates for order and n-jump comparability for all n, and constant symbols for least and greatest elements, is decidable. The decidability follows from our main theorem, where we show that any finite L-structure which is consistent with the order relation, the order-preserving property of the jump operator, and the property of the jump operator that the jump of an element is strictly greater than the element, can be embedded into the r.e. degrees. 1996Academic Press, Inc.

Computability theory, algorithmic randomness and Turing's anticipation

“…More complex versions of the infinite injury method allowed for very complex results in involving n-th jumps, partial orderings and embeddings such as Lerman-Lempp [87], and things about arithmetical definability such as Harrington [67] (See Odifreddi [111] for this). These methods have been applied by Ash and Knight in effective algebra [3], and model theory (e.g.…”

Immunity and Hyperimmunity for Sets of Minimal Indices

“…The main lemma of this section is Corollary 4.1, which follows from Lerman's revision [9] of Theorem 2.8 to account for the join operator. That the jump operator can be included when greatest element is omitted from the language was also mentioned in the discussion following [8,Theorem 7.10]. …”

“…Theorem 2.8 (Lempp and Lerman [8]). Any countable partial order P with jump which is consistent with:…”

Iterated trees of strategies and priority arguments

“…Many others have presented frameworks for priority arguments or classes of priority arguments since that time, and many new ideas have changed the way we approach priority arguments. A listing of some of these attempts can be found in [LL4]. There are several key ideas which have greatly influenced the framework which is intuitively described in this paper, and we would like to point those out.…”

“…">Introduction.It is our intent, in this paper, to try to describe some of the key ideas developed in the series or papers by Lempp and Lerman [LL1,LL2,LL3,LL4], which use the iterated trees of strategies approach to priority arguments. This is an approach which provides a framework for carrying out priority arguments at all levels of the arithmetical hierarchy.…”