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COMPUTING STRENGTH OF STRUCTURES RELATED TO THE FIELD OF REAL NUMBERS

Published online by Cambridge University Press:  21 March 2017

GREGORY IGUSA ,
JULIA F. KNIGHT  and
NOAH DAVID SCHWEBER
GREGORY IGUSA
Affiliation:
MATHEMATICS DEPARTMENT UNIVERSITY OF NOTRE DAME NOTRE DAME, IN, USA and SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON NEW ZEALAND E-mail: gregigusa@gmail.com
JULIA F. KNIGHT
Affiliation:
MATHEMATICS DEPARTMENT UNIVERSITY OF NOTRE DAME NOTRE DAME, IN, USA E-mail: knight.1@nd.edu
NOAH DAVID SCHWEBER
Affiliation:
MATHEMATICS DEPARTMENT UNIVERSITY OF CALIFORNIA BERKELEY, CA, USA and MATHEMATICS DEPARTMENT UNIVERSITY OF WISCONSIN MADISON, WI, USA E-mail: schweber@berkeley.edu
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Abstract

In [8], the third author defined a reducibility $\le _w^{\rm{*}}$ that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals ${\cal R}$ lies strictly above certain related structures. In the present paper, we show that $\left( {{\cal R},exp} \right) \equiv _w^{\rm{*}}{\cal R}$ . More generally, for the weak-looking structure ${\cal R}$ ℚ consisting of the real numbers with just the ordering and constants naming the rationals, all o-minimal expansions of ${\cal R}$ ℚ are equivalent to ${\cal R}$ . Using this, we show that for any analytic function f, $\left( {{\cal R},f} \right) \equiv _w^{\rm{*}}{\cal R}$ . (This is so even if $\left( {{\cal R},f} \right)$ is not o-minimal.)

Keywords

03C57 (also, 03C64, 03D45)generic Muchnik reducibilityordered field of realso-minimal theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 137 - 150
Copyright
Copyright © The Association for Symbolic Logic 2017 

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