On the strength of Hausdorff's gap condition

TL;DR

This paper investigates the strength of Hausdorff's gap condition by constructing an indestructible gap that does not satisfy Hausdorff's condition, using uncountably many random reals, thus resolving an open problem.

Contribution

It demonstrates that Hausdorff's gap condition is strictly weaker than the indestructibility condition by providing a novel construction.

Findings

Constructed an indestructible (ω₁,ω₁) gap not satisfying Hausdorff's condition.

Showed that Hausdorff's condition is not equivalent to indestructibility.

Used uncountably many random reals in the construction.

Abstract

Hausdorff's gap condition was satisfied by his original 1936 construction of an (omega-1,omega-1) gap in P(N)/Fin. We solve an open problem in determining whether Hausdorff's condition is actually stronger than the more modern indestructibility condition, by constructing an indestructible (omega-1,omega-1) gap not equivalent to any gap satisfying Hausdorff's condition, from uncountably many random reals.