More on HOD-supercompactness

Apter, Arthur W.; Friedman, Shoshana; Fuchs, G.

doi:10.1016/j.apal.2020.102901

Search citation statements

Order By: Relevance

Paper Sections

Select...

Proof Of (1) Assume the Contrary And Let1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

Publication Types

Select...

Article1

Relationship

Self Cite1

Independent0

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem is just an example. It is interesting to note that iterated Cohen forcing has been put to good use in the context of ordinal definability in [21] as well.…”

Section: Proof Of (1) Assume the Contrary And Letmentioning

confidence: 99%

Blurry Definability

Fuchs

2022

Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The following theorem is just an example. It is interesting to note that iterated Cohen forcing has been put to good use in the context of ordinal definability in [21] as well.…”

Section: Proof Of (1) Assume the Contrary And Letmentioning

confidence: 99%