  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.

Logic