A hierarchy of Ramsey-like cardinals

We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy

Indestructibility of Vopenka's Principle

We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to prove the relative consistency of these large cardinal axioms with a variety of statements known to be independent of ZFC, such as the generalised continuum hypothesis, the existence of a definable well-order of the universe, and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic

Consecutive singular cardinals and the continuum function

We show that from a supercompact cardinal \kappa, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, \kappa\ and \kappa^+ are both singular, and the continuum function at \kappa\ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of \kappa\ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of ZF + not AC_\omega\ in which either (1) aleph_1 and aleph_2 are both singular and the continuum function at aleph_1 can be precisely controlled, or (2) aleph_\omega\ and aleph_{\omega+1} are both singular and the continuum function at aleph_\omega\ can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals \kappa\ and \kappa^+ in a model of ZF. Some open questions concerning the continuum function in models of ZF with consecutive singular cardinals are posed.Comment: to appear in the Notre Dame Journal of Formal Logic, issue 54:3, June 201