This paper attempts to present and organize several problems in the
theory of Singular Cardinals. The most famous problems in the area
(bounds for the ℶ-function at singular cardinals) are
well known to all mathematicians with even a rudimentary interest
in set theory. However, it is less well known that the combinatorics
of singular cardinals is a thriving area with results and problems
that do not depend on a solution of the Singular Cardinals
Hypothesis. We present here an annotated collection of
representative problems with some references. Where the problems
are novel, attribution is attempted and it is noted where money is
attached to particular problems.
Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square
and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF
theory. Important subthemes were mutual stationarity, Aronszajn
trees, and superatomic Boolean Algebras.
The author notes considerable overlap between this paper and the
unpublished report submitted to the Banff Center for the Workshop
on Singular Cardinals Combinatorics, May 1–5, 2004.
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Mathematical Reviews (MathSciNet):
MR52:106
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