A framework for studying minimally inconsistent models, called ACLuN2, was formulated by Diderik Batens D. [Inconsistency-adaptive logics. In: Logic at Work, Springer, 1999, pp.445-472]. The system ACLuN2 is a central example of what are known as adaptive logics: it is taken as a point of departure for the present talk. We formulate a transfinite variant of the propositional fragment of the system ACLuN2 and investigate some of its proof-theoretic properties. We are especially interested in the property of final derivability and in the (possibly transfinite) minimal ordinal stage at which a sentence can be conclusively established from a set of premises. (This talk reports on joint work with Liza Verhoeven.)