Displaying similar documents to “Singular cardinals and strong extenders”

Singular Failures of GCH and Level by Level Equivalence

Arthur W. Apter (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is an unbounded set of singular cardinals which witness the only failures of GCH in the universe. In this model, the structure of the class of supercompact cardinals can be arbitrary.

Hybrid Prikry forcing

Dima Sinapova (2015)

Fundamenta Mathematicae

Similarity:

We present a new forcing notion combining diagonal supercompact Prikry forcing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ, and GCH holds below κ. Moreover we define a scale at κ which has a stationary set of bad points in the ground model.

More Easton theorems for level by level equivalence

Arthur W. Apter (2012)

Colloquium Mathematicae

Similarity:

We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize Theorem 1 in our earlier paper [Math. Logic Quart. 51 (2005)]. In both our ground model and the model witnessing the conclusions of our present theorems, there are no restrictions on the structure of the class of supercompact cardinals.

Level by Level Inequivalence, Strong Compactness, and GCH

Arthur W. Apter (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.

Indestructibility, strongness, and level by level equivalence

Arthur W. Apter (2003)

Fundamenta Mathematicae

Similarity:

We construct a model in which there is a strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing and in which level by level equivalence between strong compactness and supercompactness holds non-trivially.