Blurry definability

18.01.2022 15:00 - 16:30

G. Fuchs (City College of New York; US)

In this talk on ongoing research, I analyze blurry forms of ordinal definability and their hereditary versions which generalize ideas due to Hamkins/Leahy and Tzouvaras. Classically, a set is ordinal definable if it is the unique object satisfying some first order property in which ordinal parameters may occur. Given a cardinal kappa, I define that a set is \(<\kappa\)-blurrily ordinal definable if it belongs to an OD set of cardinality less than kappa. So in this case, the set is one of fewer than \(\kappa\) many objects with a certain property using ordinal parameters, not necessarily the unique such set. I will present some results on the class of hereditarily \(<\kappa\)-blurrily definable sets and the structure of leaps, that is, stages at which new sets become blurrily definable. There are some ZF(C) results and some relative consistency results using forcing.



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