Jónsson Cardinals, Erdös Cardinals, and The Core Model

On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero 4] calculates correctly the successors of JJ onsson cardinals, assuming O Sword does not exist. Namely, if is a JJ onsson cardinal then + = +K , provided that there is no non-trivial elementary embedding j : K ?! K. There are a number of related results in ZF C concerning P() in...

The Maximality of the Core Model

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K‖κ is universal for mice of height ≤ κ whenever κ ≥ א2, 5) if there is a κ such that κ is either a si...

Making All Cardinals Almost Ramsey ∗ † ‡ Arthur

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ¬ACω in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular c...

Bounded Martin's Maximum, Weak Erdös Cardinals and psi AC

Ramsey-like cardinals II

This paper continues the study of the Ramsey-like large cardinals introduced in [Git09] and [WS08]. Ramsey-like cardinals are defined by generalizing the “existence of elementary embeddings” characterization of Ramsey cardinals. A cardinal κ is Ramsey if and only if every subset of κ can be put into a κ-size transitive model of ZFC for which there exists a weakly amenable countably complete ult...