Partial strong compactness and squares

Yair Hayut
2019 Fundamenta Mathematicae  
We analyze the connection between some properties of partially strongly compact cardinals: the completion of filters of certain size and instances of the compactness of Lκ,κ. Using this equivalence we show that if any κ-complete filter on λ can be extended to a κ-complete ultrafilter and λ <κ = λ then (µ) fails for all regular µ ∈ [κ, 2 λ ]. As an application, we improve the lower bound for the consistency strength of κ-compactness, a case which was explicitly considered by Mitchell. 2010
more » ... atics Subject Classification: Primary 03E55. ( 1 ) In [8, Chapter 22], the term λ-compactness is used to denote what we call λ-strong compactness. We prefer the more cumbersome name in order to avoid inconsistency with the term κ-compact, which refers to a cardinal κ that has the κ-filter extension property. Our terminology also differs from the terminology of Bagaria and Magidor [2], where it refers to the degree of completeness of the extended ultrafilters.
doi:10.4064/fm626-9-2018 fatcat:x3azwonlv5dwjpxy2w3kozavdm