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

On the strength of Hausdorff's gap condition