On uniform weak Königʹs lemma Original Research Article

The so-called weak Königʹs 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 View the MathML source-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 View the MathML source of PRA (together with a quantifier-free axiom of choice schema which—relative to View the MathML source—implies the schema of View the MathML source-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 Königʹs lemma provided that View the MathML source 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 View the MathML source-conservative over PRA, View the MathML source 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 View the MathML source both with and without the Markov principle.