Resurrection axioms and uplifting cardinals

Notes to “The Resurrection Axioms”

I will discuss a new class of forcing axioms, the Resurrection Axioms (RA), and the Weak Resurrection Axioms (wRA). While Cohen’s method of forcing has been designed to change truths about the set-theoretic universe you live in, the point of Resurrection is that some truths that have been changed by forcing can in fact be resurrected, i.e. forced to hold again. In this talk, I will illustrate h...

Large cardinals and resurrection axioms

The Two-cardinals Transfer Property and Resurrection of Supercompactness

We show that the transfer property (א1,א0)→ (λ+, λ) for singular λ does not imply (even) the existence of a non-reflecting stationary subset of λ+. The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals; to show that ...

The Wholeness Axioms and the Class of Supercompact Cardinals

We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.

Ramsey-like cardinals

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. I introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. T...