Math 655: Set Theory (Spring 2019)
Class Meets MWF 9:30–10:20
Room Keller 403
Office PSB 305
Office Hours MWTh 10:30–11:30
Email kamerynw [at] hawaii [period] edu
Large Cardinals and Forcing
The unofficial title for this course is “Large cardinals and forcing”. As the title suggests, I hope to introduce you to two of the major themes of contemporary set theoretic research—large cardinals and forcing—and the interaction between the two.
Large cardinals are those cardinal numbers which are “too big” for the standard axioms of set theory, ZFC, to prove their existence. A variety of large cardinal properties have been considered, starting with work of Hausdorff, Ulam, and others. The large cardinals give rise to a hierarchy of theories which extend ZFC in logical strength, giving us a yardstick by which to measure the strength of principles which exceed ZFC.
Forcing was originally developed by Paul Cohen to prove the consistency of the failure of the continuum hypothesis. It is a remarkably flexible method and is the main tool used to prove consistency and independence results.
Due to time constraints, there are many topics we will not be able to touch on, among them inner models, determinancy, descriptive set theory, and infinitary combinatorics.
I will not collect homework for grading, but you are encouraged to work on the exercises I mention in class, and talk to me about your solutions or attempted solutions.
As part of the course, I will ask each student to confer with me to pick a topic to research. Everyone will be asked to give a short presentation about their topic at some point during the semester. There are myriad possible topics, and I will give suggestions along the way.
Tentative Course Outline

Part 0: Introduction (~3 weeks)

Cantorian set theory (ordinals and cardinals, transfinite recursion)

Axioms of set theory

The cumulative hierarchy and reflection

Small large cardinals


Part 1: Large Cardinals (~4 weeks)

Inaccessible cardinals

Measurable cardinals

Weakly compact cardinals

Supercompact cardinals


Part 2: Forcing (~4 weeks)

The forcing relation and the forcing theorem

The independence of the continuum hypothesis

Product and iterated forcing

Some cardinal characteristics of the continuum


Part 3: Large Cardinals and Forcing (~3 weeks)

Some subset of the following topics:

Large cardinals cannot settle the continuum hypothesis

Prikry forcing and the singular cardinals hypothesis

The number of normal measures on $\kappa$

The tree property on $\omega_2$

Possibly others…

Projects
As part of this course I will ask each of you to research a topic. This topic is to be related in some way to set theory, and chosen with my approval. I will ask of you to give a short in class presentation about your topic toward the end of the semester.
Some possible areas for topics:

Determinacy of infinite games

Set theoretic topology

Cardinal arithmetic without the axiom of choice

Infinite time/space Turing machines

Higher recursion theory/admissible set theory

Infinite Ramsey theory

Secondorder set theory
Textbook
The “official” textbook for the course is Kunen’s Set Theory (ISBN: 9781848900509). As supplementary books I will reference Kanamori’s The Higher Infinite (ISBN: 3540003843) and Jech’s Set Theory, 3rd ed. (ISBN:3540440852). In particular, Kunen says little about large cardinals and Kanamori will be the primary reference for Part 1.
I will strive to regularly post lecture notes, in addition to what can be found in the texts.
Grading Policy
I anticipate everyone signed up for this class getting an A. You should show up and participate in class, present your project, and spend some time on the exercises. I reserve the right to assign a lower grade for exceptional cases. For really exceptional cases, see this Futurama clip.
Announcements
(Nothing yet.)
Lecture notes
These notes are licensed under a Creative Commons AttributionNonCommercialShareAlike 4.0 International License.