Notre Dame Journal of Formal Logic

Primitive Recursion and the Chain Antichain Principle

Alexander P. Kreuzer


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 $\text{RT}^2_2$.

Article information

Notre Dame J. Formal Logic, Volume 53, Number 2 (2012), 245-265.

First available in Project Euclid: 9 May 2012

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Zentralblatt MATH identifier

Primary: 03F35: Second- and higher-order arithmetic and fragments [See also 03B30] 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F10: Functionals in proof theory

proof mining chain antichain principle conservation bar recursion Erdos-Moser principle


Kreuzer, Alexander P. Primitive Recursion and the Chain Antichain Principle. Notre Dame J. Formal Logic 53 (2012), no. 2, 245--265. doi:10.1215/00294527-1715716.

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