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COMPARING THE STRENGTH OF DIAGONALLY NONRECURSIVE FUNCTIONS IN THE ABSENCE OF ${\rm{\Sigma }}_2^0$ INDUCTION

Published online by Cambridge University Press:  22 December 2015

FRANÇOIS G. DORAIS ,
JEFFRY L. HIRST  and
PAUL SHAFER
FRANÇOIS G. DORAIS
Affiliation:
DEPARTMENT OF MATHEMATICS DARTMOUTH COLLEGE HANOVER NH 03755, USAE-mail: francois.g.dorais@dartmouth.eduURL: http://math.dartmouth.edu/∼dorais/
JEFFRY L. HIRST
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES APPALACHIAN STATE UNIVERSITY BOONE NC 28608, USAE-mail: jlh@math.appstate.eduURL: http://mathsci2.appstate.edu/∼jlh/
PAUL SHAFER
Affiliation:
DEPARTMENT OF MATHEMATICS GHENT UNIVERSITY KRIJGSLAAN 281 S22 B-9000 GHENT, BELGIUME-mail: paul.shafer@ugent.beURL: http://cage.ugent.be/∼pshafer/
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Abstract

We prove that the statement “there is a k such that for every f there is a k-bounded diagonally nonrecursive function relative to f” does not imply weak König’s lemma over ${\rm{RC}}{{\rm{A}}_0} + {\rm{B\Sigma }}_2^0$. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally nonrecursive function computes a 2-bounded diagonally nonrecursive function may fail in the absence of ${\rm{I\Sigma }}_2^0$.

Keywords

Reverse mathematicsdiagonally nonrecursive functionsWKLDNRnonstandard models of arithmeticgraph coloring
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1211 - 1235
Copyright
Copyright © The Association for Symbolic Logic 2015 

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