Blurry Definability

Gunter Fuchs
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Gunter Fuchs: Department of Mathematics, College of Staten Island (CUNY), 2800 Victory Blvd., Staten Island, New York, NY 10314, USA

Mathematics, 2022, vol. 10, issue 3, 1-34

Abstract: I begin the study of a hierarchy of (hereditarily) < κ -blurrily ordinal definable sets. Here for a cardinal κ , a set is < κ -blurrily ordinal definable if it belongs to an OD set of cardinality less than κ , and it is hereditarily so if it and each member of its transitive closure is. I show that the class of hereditarily < κ -blurrily ordinal definable sets is an inner model of ZF . It satisfies the axiom of choice iff it is a κ -c.c. forcing extension of HOD , and HOD is definable inside it (even if it fails to satisfy the axiom of choice). Of particular interest are cardinals λ such that some set is hereditarily < λ -blurrily ordinal definable but not hereditarily < κ -blurrily ordinal definable for any cardinal κ < λ . Such cardinals I call leaps. The main results concern the structure of leaps. For example, I show that if λ is a limit of leaps, then the collection of all hereditarily < λ -blurrily ordinal definable sets is a model of ZF in which the axiom of choice fails. Using forcing, I produce models exhibiting various leap constellations, for example models in which there is a (regular/singular) limit leap whose cardinal successor is a leap. Many open questions remain.

Keywords: ordinal definability; HOD; forcing; axiom of choice (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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