Finding paths through narrow and wide trees

TL;DR

This paper compares the logical strength of axioms asserting the existence of infinite paths in narrow and wide trees within second-order arithmetic, showing their relative weakness and incomparability to established principles like Weak König's Lemma.

Contribution

It introduces and analyzes two axioms about paths in narrow and wide trees, establishing their position in the hierarchy of second-order arithmetic principles.

Findings

Both axioms are weaker than Weak König's Lemma.

The axioms are incomparable in strength to WWKL.

The results clarify the logical landscape of tree path principles.

Abstract

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak K\"onig's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.