### Cofinality spectrum problems: the axiomatic approach

by Malliaris and Shelah. [MiSh:1070]

Topology and its Applications, 2016

Let X be a set of definable linear or partial orders in some
model. We say that X has emph {uniqueness} below some cardinal
{t}_* if for any regular kappa < {t}_*, any
two increasing kappa-indexed sequences in any two orders of Xhave the same co-initiality. Motivated by recent work, we investigate
this phenomenon from several interrelated points of view.
We define the lower-cofinality spectrum for a regular ultrafilter
{D} on lambda and show that this spectrum may consist of
a strict initial segment of cardinals below lambda and also that
it
may finitely alternate. We connect these investigations to a
question
of Dow on the conjecturally nonempty (in ZFC) region of OK but
not
good ultrafilters, by introducing the study of so-called `automorphic
ultrafilters' and proving that the ultrafilters which are automorphic
for some, equivalently every, unstable theory are precisely the
good
ultrafilters. Finally, we axiomatize a general context of
``lower cofinality spectrum problems'', a bare-bones framework
consisting essentially of a single tree projecting onto two linear
orders. We prove existence of a lower cofinality function in
this
context show that this framework holds of theories which are
substantially less complicated than Peano arithmetic, the natural
home of cofinality spectrum problems. Along the way we give
new analogues of several open problems.

Back to the list of publications