×

Reverse mathematics and a Ramsey-type König’s lemma. (English) Zbl 1259.03022

In this paper a new principle, \(\mathsf{RKL}\), is introduced and its strength, in the sense of reverse mathematics, is studied. \(\mathsf{RKL}\) states that for an infinite binary tree \(T\subseteq 2^{<\mathbb N}\) there is a set \(H\) which is homogeneous in the sense that there are arbitrarily long \(\sigma\in T\) with \(\sigma(x)=c\) for all \(x\in H\) with \(x< | \sigma|\). (So in some sense the tree represents a coloring.) It is shown that \(\mathsf{RKL}\) is a consequence of \(\mathsf{WKL}\) and Ramsey’s theorem for pairs, \(\mathsf{RT}^2_2\) (more precisely \(\mathsf{SRT}^2_2\)). Moreover, the author studies variants of this principle.

MSC:

03B30 Foundations of classical theories (including reverse mathematics)
03F35 Second- and higher-order arithmetic and fragments
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] DOI: 10.1090/S0002-9939-09-10115-6 · Zbl 1195.03015
[2] Corrigendum to: ”On the strength of Ramsey’s theorem for pairs” 74 pp 1438– (2009)
[3] On the strength of Ramsey’s theorem for pairs 66 pp 1– (2001) · Zbl 0977.03033
[4] Recursively enumerable sets and degrees (1987)
[5] Subsystems of second order arithmetic (2009) · Zbl 1181.03001
[6] DOI: 10.1305/ndjfl/1040136917 · Zbl 0843.03034
[7] A set with no infinite low subset in either it or its complement 66 pp 1371– (2001) · Zbl 0990.03046
[8] DOI: 10.1002/malq.19930390153 · Zbl 0799.03048
[9] classes and Boolean combinations of recursively enumerable sets 39 pp 95– (1974)
[10] DOI: 10.1142/9789812796554_0008
[11] DOI: 10.1215/00294527-2010-039 · Zbl 1217.03019
[12] Effective presentability of Boolean algebras of Cantor-Bendixson rank 1 64 pp 45– (1999) · Zbl 0924.03084
[13] does not imply WKL 77 pp 609– (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.