On the Uniform Weak König's Lemma

The so-called weak König's lemma WKL asserts the existence of an infinite path

in any infinite binary tree (given by a representing function

). 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 $\Pi^0_2$ -conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In 1992 we established such conservation results relative to finite type extensions PRA $^{\omega}$ of PRA (together with a quantifier-free axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional $\Phi$ which selects uniformly in a given infinite binary tree

an infinite path $\Phi f$ 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 from 1992 actually can be used to eliminate even this uniform weak König's lemma provided that PRA $^{\omega}$ only has a quantifier-free rule of extensionality QF-ER instead of the full axioms

of extensionality for all finite types. In this paper we show that in the presence of

, UWKL is much stronger than WKL: whereas WKL remains to be $\Pi^0_2$ -conservative over PRA, PRA $^{\omega}+(E)+$ UWKL contains (and is conservative over) full Peano arithmetic PA

Abstract: