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FINDING DESCENDING SEQUENCES THROUGH ILL-FOUNDED LINEAR ORDERS

Published online by Cambridge University Press:  01 February 2021

JUN LE GOH
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSINMADISON, WI, USAE-mail:junle.goh@wisc.edu
ARNO PAULY
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF SWANSEASWANSEA, UKE-mail:arno.m.pauly@gmail.com
MANLIO VALENTI
Affiliation:
DEPARTMENT OF MATHEMATICS, COMPUTER SCIENCE AND PHYSICS UNIVERSITY OF UDINEUDINE, ITALYE-mail:manliovalenti@gmail.com

Abstract

In this work we investigate the Weihrauch degree of the problem Decreasing Sequence ( $\mathsf {DS}$ ) of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem Bad Sequence ( $\mathsf {BS}$ ) of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf {DS}$ , despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$ , $\boldsymbol {\Sigma }^1_1$ , $\boldsymbol {\Pi }^1_1$ . We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.

Type
Article
Copyright
© The Association for Symbolic Logic 2021

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