We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ-Erdős in K. In the absence of the Steel core model K we prove the same conclusion for any model L[E ] such that either V = L[E ] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardina...

A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We ...

Cardinal direction relations are binary spatial relations determined under an extrinsically-defined direction system (e.g., north of). We already have point-based and region-based cardinal direction calculi, but for the relations between other combinations of objects we have only a model. We are, therefore, developing heterogeneous cardinal direction calculus, which allows reasoning on cardinal...

The purpose of this study is to reduce the uncertainty in calculation process on hesitant fuzzy sets (HFSs). innovation unify cardinal numbers elements (HFEs) a special way. Firstly, probability density function assigned for any given HFE. Thereafter, equal-probability transformation introduced transform HFEs with different condition into same function. characteristic that higher consistency me...