MoL-2013-12: Determinacy and measurable cardinals in HOD

MoL-2013-12: Tzimoulis, Apostolos (2013) Determinacy and measurable cardinals in HOD. [Report]

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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.

Item Type: Report
Report Nr: MoL-2013-12
Series Name: Master of Logic Thesis (MoL) Series
Year: 2013
Uncontrolled Keywords: Logic; Mathematics
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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