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EXPANDING THE REALS BY CONTINUOUS FUNCTIONS ADDS NO COMPUTATIONAL POWER

Part of: Set theory Computability and recursion theory

Published online by Cambridge University Press:  26 September 2022

URI ANDREWS ,
JULIA F. KNIGHT ,
RUTGER KUYPER ,
JOSEPH S. MILLER Open the ORCID record for JOSEPH S. MILLER [Opens in a new window]  and
MARIYA I. SOSKOVA Open the ORCID record for MARIYA I. SOSKOVA [Opens in a new window]
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON, MADISON, WI, USA E-mail: andrews@math.wisc.edu E-mail: jmiller@math.wisc.edu
JULIA F. KNIGHT
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME, SOUTH BEND, IN, USA E-mail: Julia.F.Knight.1@nd.edu
RUTGER KUYPER
Affiliation:
METASWITCH WELLINGTON, NEW ZEALAND E-mail: mail@rutgerkuyper.com
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON, MADISON, WI, USA E-mail: andrews@math.wisc.edu E-mail: jmiller@math.wisc.edu
MARIYA I. SOSKOVA*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–MADISON, MADISON, WI, USA E-mail: andrews@math.wisc.edu E-mail: jmiller@math.wisc.edu
*
E-mail: msoskova@math.wisc.edu
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Abstract

We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any Borel quotient of the reals reduces to it.

Keywords

generic Muchnik reducibilityuncountable structurescontinuous functionsordered field of the realsrunning jump

MSC classification

Primary: 03D30: Other degrees and reducibilities
Secondary: 03D45: Theory of numerations, effectively presented structures 03E40: Other aspects of forcing and Boolean-valued models
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 3 , September 2023 , pp. 1083 - 1102
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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