On uniform weak König's lemma

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Abstract

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree (given by a representing function f). Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π20-conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In Kohlenbach [10] (J. Symbolic Logic 57 (1992) 1239–1273) we established such conservation results relative to finite type extensions PRAω of PRA (together with a quantifier-free axiom of choice schema which—relative to PRAω—implies the schema of Σ10-induction). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRAω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms (E) of extensionality for all finite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be Π20-conservative over PRA, PRAω+(E)+UWKL contains (and is conservative over) full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRAω both with and without the Markov principle.

MSC

03F10
03F25
03F35
03F50
03D65

Keywords

König's lemma
Higher order arithmetic
Functionals of finite type
Fan rule
Explicit mathematics

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