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 yard-stick 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

• Second-order set theory

## Textbook

The “official” textbook for the course is Kunen’s Set Theory (ISBN: 978-1-84890-050-9). As supplementary books I will reference Kanamori’s The Higher Infinite (ISBN: 3-540-00384-3) and Jech’s Set Theory, 3rd ed. (ISBN:3-540-44085-2). 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.