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COMPARING TWO VERSIONS OF THE REALS

Published online by Cambridge University Press:  12 August 2016

G. IGUSA  and
J. F. KNIGHT
G. IGUSA
Affiliation:
UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556, USAE-mail: gigusa@nd.edu
J. F. KNIGHT
Affiliation:
UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556, USAE-mail: knight.1@nd.edu
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Abstract

Schweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$. We prove that ${\cal R}$ is not reducible to ${\cal W}$. As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

Keywords

computabilityuncountable structures
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 1115 - 1123
Copyright
Copyright © The Association for Symbolic Logic 2016 

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