In this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations, but which require that the "density" of these errors be no more than a certain, prespectified amount. We introduce three hierarchies of such criteria, each of which has the same order type as the real unit interval. These three hierarchies are progressively more strict in the way they measure density of errors of explanations. The strictest of the three turns out to have all of its members, save one, incomparable to the identification criterion which permits finitely many errors in explanations.
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