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RAMSEY’S THEOREM FOR PAIRS AND K COLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

Published online by Cambridge University Press:  19 June 2017

STEFANO BERARDI  and
SILVIA STEILA
STEFANO BERARDI
Affiliation:
DIPARTIMENTO DI INFORMATICA UNIVERSITÀ DEGLI STUDI DI TORINO CORSO SVIZZERA 185 10149 TORINO, ITALYE-mail: stefano@di.unito.it
SILVIA STEILA
Affiliation:
INSTITUTE OF COMPUTER SCIENCE UNIVERSITY OF BERN NEUBRÜCKSTRASSE 10 CH-3012 BERN, SWITZERLANDE-mail: steila@inf.unibe.ch
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Abstract

The purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments of k-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for ${\rm{\Sigma }}_3^0$ formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite k-ary tree there is some $i < k$ and some branch with infinitely many children of index i.

Keywords

03F5503B30Ramsey’s Theoremintuitionistic arithmeticPrinciple of Omniscience

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 737 - 753
Copyright
Copyright © The Association for Symbolic Logic 2017 

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