MoL-2013-12:
Tzimoulis, Apostolos
(2013)
*Determinacy and measurable cardinals in HOD.*
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## 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.

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 |

URI: | https://eprints.illc.uva.nl/id/eprint/901 |

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