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2012 Primitive Recursion and the Chain Antichain Principle
Alexander P. Kreuzer
Notre Dame J. Formal Logic 53(2): 245-265 (2012). DOI: 10.1215/00294527-1715716

Abstract

Let the chain antichain principle (CAC) be the statement that each partial order on $\mathbb{N}$ possesses an infinite chain or an infinite antichain. Chong, Slaman, and Yang recently proved using forcing over nonstandard models of arithmetic that CAC is $\Pi^1_1$-conservative over $\text{RCA}_0+\Pi^0_1\text{-CP}$ and so in particular that CAC does not imply $\Sigma^0_2$-induction. We provide here a different purely syntactical and constructive proof of the statement that CAC (even together with WKL) does not imply $\Sigma^0_2$-induction. In detail we show using a refinement of Howard's ordinal analysis of bar recursion that $\text{WKL}_0^\omega+\text{CAC}$ is $\Pi^0_2$-conservative over PRA and that one can extract primitive recursive realizers for such statements. Moreover, our proof is finitary in the sense of Hilbert's program. CAC implies that every sequence of $\mathbb{R}$ has a monotone subsequence. This Bolzano-Weierstraß}-like principle is commonly used in proofs. Our result makes it possible to extract primitive recursive terms from such proofs. We also discuss the Erdős-Moser principle, which—taken together with CAC—is equivalent to $\text{RT}^2_2$.

Citation

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Alexander P. Kreuzer. "Primitive Recursion and the Chain Antichain Principle." Notre Dame J. Formal Logic 53 (2) 245 - 265, 2012. https://doi.org/10.1215/00294527-1715716

Information

Published: 2012
First available in Project Euclid: 9 May 2012

zbMATH: 1253.03090
MathSciNet: MR2925280
Digital Object Identifier: 10.1215/00294527-1715716

Subjects:
Primary: 03B30, 03F10, 03F35

Rights: Copyright © 2012 University of Notre Dame

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Vol.53 • No. 2 • 2012
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