We give a set-valued criterion for a topological space $X$ to be consonant, i.e., the upper Kuratowski topology on the family of all closed subsets of $X$ coincides with the co-compact topology. This characterization of consonance is then used to show that the statement `every analytic metrizable consonant space is complete' is independent of the usual axioms of set theory. This answers a question by Nogura and Shakhmatov. It is also proved that continuous open surjections defined on a consonant space are compact covering.