The Strength of Menger's Conjecture

TL;DR

This paper investigates Menger's conjecture regarding the σ-compactness of subsets of R with the Menger property, showing that for projective sets, the conjecture's consistency strength is only an inaccessible cardinal.

Contribution

The paper proves that Menger's conjecture for projective sets has a lower consistency strength than previously thought, specifically only requiring an inaccessible cardinal.

Findings

Menger's conjecture for projective sets is consistent with an inaccessible cardinal.

The conjecture's validity depends on large cardinal assumptions.

The consistency strength is lower than that implied by the Axiom of Projective Determinacy.

Abstract

Menger conjectured that subsets of R with the Menger property must be ${\sigma}$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We note that in fact, Menger's conjecture for projective sets has consistency strength of only an inaccessible cardinal.