Translator Disclaimer
December 2008 Π⁰₁ classes with complex elements
Stephen Binns
J. Symbolic Logic 73(4): 1341-1353 (December 2008). DOI: 10.2178/jsl/1230396923

Abstract

An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π₁⁰ class P contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every Y⊆ω there is an X in P such that X≥wtt Y. We show that this is also equivalent to the Π₁⁰ class's being large in some sense. We give an example of how this result can be used in the study of scattered linear orders.

Citation

Download Citation

Stephen Binns. "Π⁰₁ classes with complex elements." J. Symbolic Logic 73 (4) 1341 - 1353, December 2008. https://doi.org/10.2178/jsl/1230396923

Information

Published: December 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1155.03032
MathSciNet: MR2467221
Digital Object Identifier: 10.2178/jsl/1230396923

Subjects:
Primary: computability theory , recursion theory

Keywords: complex , computably growing , Muchnik , randomness , ranked , scattered linear order , Π⁰₁ class

Rights: Copyright © 2008 Association for Symbolic Logic

JOURNAL ARTICLE
13 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.73 • No. 4 • December 2008
Back to Top