Notions of Strong Compactness without the Axiom of Choice
Vincent Kieftenbeld

Abstract:
The study of large cardinal axioms is an active part of contemporary
set theory. For a large cardinal notion there are often several
definitions possible. For example, two common ways to define a large
cardinal notion is as a critical point of an elementary embedding with
certain properties, or in terms of ultrafilters.  Many other types of
definitions exist. With the axiom of choice these definitions are
often equivalent. Without the axiom of choice, these definitions may
not be equivalent anymore. Moreover the consistency strength of the
large cardinal axiom may change with the ambient set theory, depending
on which definition you choose.  In this thesis we study several
different definitions related to the notion of a compact cardinal. We
will be guided by two main questions: What is the structure of
implications between different definitions? And: What is the relative
consistency strength of these definitions? In both cases the answers
may depend on the presence or absence of the axiom of choice.