1 Department of Computer Science, Faculty of Science, Aarhus University, Aarhus University2 unknown
The so-called weak Konig'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 20-conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In Kohlenbach  (J. Symbolic Logic 57 (1992) 1239-1273) we established such conservation results relative to finite type extensions PRA of PRA (together with a quantifier-free axiom of choice schema which-relative to PRA -implies the schema of 10-induction). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional which selects uniformly in a given infinite binary tree f an infinite path f 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  actually can be used to eliminate even this uniform weak Konig's lemma provided that PRA 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 20-conservative over PRA, PRA +(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 PRA both with and without the Markov principle.
Annals of Pure and Applied Logic, 2002, Vol 114, Issue 1, p. 103-116
Konig's lemma; Higher order arithmetic; Functionals of finite type; Fan rule; Explicit mathematics