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Fall 1995 On the Strength of Ramsey's Theorem
David Seetapun, Theodore A. Slaman
Notre Dame J. Formal Logic 36(4): 570-582 (Fall 1995). DOI: 10.1305/ndjfl/1040136917

Abstract

We show that, for every partition F of the pairs of natural numbers and for every set C, if C is not recursive in F then there is an infinite set H, such that H is homogeneous for F and C is not recursive in H. We conclude that the formal statement of Ramsey's Theorem for Pairs is not strong enough to prove $ACA_0$, the comprehension scheme for arithmetical formulas, within the base theory $RCA_0$, the comprehension scheme for recursive formulas. We also show that Ramsey's Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within $RCA_0$. In particular, Ramsey's Theorem for Pairs is not conservative over $RCA_0$ for $\Pi^0_4$-sentences.

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David Seetapun. Theodore A. Slaman. "On the Strength of Ramsey's Theorem." Notre Dame J. Formal Logic 36 (4) 570 - 582, Fall 1995. https://doi.org/10.1305/ndjfl/1040136917

Information

Published: Fall 1995
First available in Project Euclid: 17 December 2002

zbMATH: 0843.03034
MathSciNet: MR1368468
Digital Object Identifier: 10.1305/ndjfl/1040136917

Subjects:
Primary: 03F35
Secondary: 03C62

Rights: Copyright © 1995 University of Notre Dame

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Vol.36 • No. 4 • Fall 1995
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