On the Complexity of the Successivity Relation in Computable Linear Orderings

Abstract

Abstract. In this paper, we solve a long-standing open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing ∆ 0 3 -isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.

On computable self-embeddings of computable linear orderings

“…The key idea of our proof is similar to the proof technique in Downey, Lempp, and G. Wu [6], who prove that for any computable linear ordering with infinitely many successivities, there is a computable isomorphic copy in which the successivity relation has degree 0'. The idea there was not to try to produce a A^-isomorphism by effectively approximating it by finite partial isomorphisms, but to define finite parts of a Aj-isomorphism along the true path of an infinite-injury priority argument on a tree of strategies.…”

Degree Spectra for Transcendence in Fields

“…Our first example shows that, for a fixed computably enumerable Turing degree c, the relations T and I can both be intrinsically of degree c. This term was used in [2], in which Downey and Moses showed that the relation of adjacency in a computable linear order can be intrinsically of degree 0 ′ . Subsequently, Downey, Lempp, and Wu showed in [1] that the only degrees c for which the adjacency relation can be intrinsically of degree c are c = 0 ′ and (if the adjacency relation is finite) c = 0. Therefore Theorem 2 distinguishes the situation for transcendence and for independence in fields from that of adjacency in linear orders.…”

Corrigendum: "On the complexity of the successivity relation in computable linear orderings"

“…Let us first start with the errors in the proof of our Main Theorem in [2] in the case of a linear order A in which any infinite interval is η-like but not strongly η-like. Specifically, in the description of the full W a -strategy in Sec.…”

“…With these two changes, Lemmas 3.2(4)(b) and 3.2(5)(c) are now correct as stated. Now let us turn to the missing case to establish our Main Theorem in [2], namely, the case when the linear order A is η-like but has no infinite strongly η-like interval nor a limit point of successivities handledà la Chubb et al It is not hard to check that any such order must contain an infinite interval of the form…”

“…For our proof below, we will actually only need the property that for all i ∈ ω, m i > 1, or for all i ∈ ω, n i > 1; the important property of such a linear order A for us is that given any a ∈ A, exactly one of the intervals (−∞, a) or (a, ∞) contains infinitely many successivities, and so we can guess at the location of the finitely many successivities in the other interval. As in the construction of our paper [2], we will build a computable copy B of A as well as, given a c.e. set C ≥ T Succ(A), a computable functional Γ with Γ Succ(B) = C.…”