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In joint work with Jouko Väänänen we discuss the relationship between the predicate representing the class of all infinite Cardinals, and that of the graph of the power set function. This work relates to the notion of inner models defined à la Gödel but using definability in languages stronger than first order which determined, L, the constructible sets.
https://www.math.uni-bonn.de/ag/logik/oberseminar-SoSe25.shtml