ON THE DEFINABILITY OF THE DOUBLE JUMP IN THE COMPUTABLY ENUMERABLE SETS-Reference-Cited by-同舟云学术

ON THE DEFINABILITY OF THE DOUBLE JUMP IN THE COMPUTABLY ENUMERABLE SETS

Published:2002-11 Issue:02 Volume:02 Page:261-296
ISSN:0219-0613
Container-title:Journal of Mathematical Logic
language:en
Short-container-title:J. Math. Log.

Author:

CHOLAK PETER A.¹,HARRINGTON LEO A.²

Affiliation:

1. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5641, USA

2. Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

Abstract

We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let [Formula: see text] is the Turing degree of a [Formula: see text] set J ≥T0″}. Let [Formula: see text] such that [Formula: see text] is upward closed in [Formula: see text]. Then there is an ℒ(A) property [Formula: see text] such that [Formula: see text] if and only if there is an A where A ≡T F and [Formula: see text]. A corollary of this is that, for all n ≥ 2, the high n ([Formula: see text]) computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin's Invariance Conjecture.

Publisher

World Scientific Pub Co Pte Lt

Subject

Logic

Link

https://www.worldscientific.com/doi/pdf/10.1142/S0219061302000151

Reference10 articles.

1. Coding in the Partial Order of Enumerable Sets

2. Definability, Automorphisms, and Dynamic Properties of Computably Enumerable Sets

3. The Δ₃⁰-automorphism method and noninvariant classes of degrees

4. Degrees of recursively enumerable sets which have no maximal supersets

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