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THE DEFINABILITY STRENGTH OF COMBINATORIAL PRINCIPLES

Published online by Cambridge University Press:  01 December 2016

WEI WANG
WEI WANG*
Affiliation:
INSTITUTE OF LOGIC AND COGNITION AND DEPARTMENT OF PHILOSOPHY SUN YAT-SEN UNIVERSITY, 135 XINGANG XI ROAD GUANGZHOU 510275, P.R. CHINAE-mail: wwang.cn@gmail.com
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Abstract

We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey’s Theorem for colorings of pairs could help in simplifying the definitions of some ${\rm{\Delta }}_2^0$ sets, while some others could not. We also investigate some consequences of Ramsey’s Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.

Keywords

03B3003F35reverse mathematicsdefinability strengthRamsey’s theoremfree setthin setrainbow Ramsey theorem
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1531 - 1554
Copyright
Copyright © The Association for Symbolic Logic 2016 

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