32,025 research outputs found
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
The singular world of singular cardinals
The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
Singular cardinals and strong extenders
We investigate the circumstances under which there exist a singular cardinal
and a short -extender witnessing " is
-strong", such that is singular in \Ult(V, E).Comment: 8 page
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