Subcompact cardinals, squares, and stationary reflection

We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by alpha^+-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.Comment: 18 pages. Corrections and improvements from referee's report mad

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

Eastonʼs theorem and large cardinals from the optimal hypothesis

AbstractThe equiconsistency of a measurable cardinal with Mitchell order o(κ)=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell (1984) [13] and M. Gitik (1989) [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see Gitik, 1993 [8]).In Friedman and Honzik (2008) [5], we formulated and proved Eastonʼs (1970) theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik (we used the assumption that the relevant target model contains H(μ), for a suitable μ, instead of the cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik (2008) [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals