More conservativity for weak Kőnig’s lemma

Abstract

We prove conservativity results for weak Kőnig’s lemma that extend the celebrated result of Harrington (for ${\Pi }_1^1$-statements) and are somewhat orthogonal to the extension by Simpson, Tanaka and Yamazaki (for statements of the form $\forall X\exists !Y\psi$ with arithmetical $\psi$). In particular, we show that ${\mathsf{WKL}}_0$ is conservative over ${\mathsf{RCA}}_0$ for well-ordering principles. We also show that compactness (which characterizes weak Kőnig’s lemma) is dispensable for certain results about continuous functions with isolated singularities.