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Finding paths through narrow and wide trees. (English) Zbl 1161.03035

The authors introduce two statements which are restrictions of Weak König’s Lemma to subclasses of the \(0\)-\(1\) trees. A node \(\sigma\) in a tree \(T\) is called extendible if the set of its extensions is infinite. It is called a branching node if both \(\sigma 0\) and \(\sigma 1\) are extendible. The axiom DIM asserts that if \(T\) is an infinite \(0\)-\(1\) tree and there is no function \(f\) such that every extendible \(\sigma\) has a branching extension of length no more than \(f(|\sigma |)\), then \(T\) has an infinite path. Since branching nodes are rare in such trees, we can think of DIM as a restriction of WKL\(_0\) to narrow trees.
The statement VSMALL is a restriction of WKL\(_0\) to a smaller subclass of narrow trees. Working over RCA\(_0\), the base system for the program of reverse mathematics, the authors show that both DIM and VSMALL are independent of both WWKL (Weak Weak König’s Lemma) and DNR. More about the relationship between WWKL and DNR can be found in [K. Ambos-Spies, B. Kjos-Hanssen, S. Lempp and T. A. Slaman, “Comparing DNR and WWKL”, J. Symb. Log. 69, No. 4, 1089–1104 (2004; Zbl 1076.03039)], for example.

MSC:

03F35 Second- and higher-order arithmetic and fragments
03B30 Foundations of classical theories (including reverse mathematics)

Citations:

Zbl 1076.03039
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References:

[1] DOI: 10.1007/s001530100100 · Zbl 1030.03044
[2] Comparing DNR and WWKL 69 pp 1089– (2004) · Zbl 1076.03039
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[11] DOI: 10.1090/S0002-9947-1968-0227009-1
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