Determinacy and measurable cardinals in HOD
Apostolos Tzimoulis
Abstract:
The study of the axiom of choice, AC, and of the axiom of determinacy,
AD, are often seen as complementary endeavours in set theory since
these axioms are incompatible. However, the contemporary development
of set theory has allowed the emergence of an intricate connection
between determinacy axioms and large cardinal axioms. In particular
the hierarchy of the consistency strength of ZFC with large cardinal
axioms has been used to gauge with precision the consistency strength
of determinacy axioms. This enterprise is twofold. On one hand large
cardinal assumptions in ZFC have been used to derive various degrees
of determinacy of projective pointclasses, as well as the consistency
of AD. On the other hand, models of AD, where AC is absent, have been
used to create inner models that satisfy AC and contain large
cardinals notions, even those that may not provably exist in a model
of AD.
In this thesis, we study the underlying technique with which some of
the latter results are achieved. Namely, taking a combinatorial large
cardinal property created in L(ωω) via the axiom of determinacy and
then pulling it back into HOD, which satisfies ZFC, resulting a much
stronger large cardinal property.
The phrase combinatorial large cardinal property is used to highlight
a difference between large cardinal properties in models of ZFC and of
ZF + AD. In ZFC, the existence of a κ-complete non-principal
ultrafilter over κ is equivalent to the existence of a non-trivial
elementary embedding with critical point κ. In ZF though, we cannot
prove this equivalence: The existence of a non-trivial elementary
embedding with critical point κ implies that κ is a large cardinal in
a meaningful way even in models of ZF + AD whereas it is consistent
with ZF that א1 carries a non-principal ω1-complete ultrafilter. In
fact, in a model of ZF + AD this is the case. We refer to the first
description of a large cardinal notion as a combinatorial notion and
the second as an embedding notion. In ZFC, the combinatorial notions
are generally equivalent to appropriate embedding notions. At the same
time, in ZF without choice, the embedding notions can be considerably
stronger than the combinatorial notions, as has been studied in
[Kie06] for example.
Here, we will first present large cardinal notions, focusing on
combinatorial and embedding formulations of measurable cardinals, and
study the relations of these with and without AC. Then, working in a
model of AD, we will show the existence of combinatorial large
cardinals. Finally we will present the technique of pulling the
combinatorial objects in HOD in order to obtain embedding large
cardinals.
Our main goal is to isolate the technique of pulling back
combinatorial properties from the models of AD to get embedding
properties in inner models that satisfy AC. This technique is not new:
it is the backbone of Woodin’s Theorem and has been used by other
authors. However, the technique has never been presented in isolation,
independent of a particular application. By focusing on large cardinal
properties that are much weaker than Woodinness, we manage to present
the technique in its purest form, allowing for easily accessible
proofs.