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.

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

• This mathoverflow answer by Joel David Hamkins does a nice job of explaining why the choice of support matters for product/iterated forcing.

• Here is the schedule for the presentations:

• Fri, April 26

• Jake Fennick, Stop Quine’ing about ‘New’ Foundations
• Mon, April 29

• Sam Birns, Infinite Time Turing Machines

• Ikenna Nometa, Infinite Ramsey’s Theorem

• Umar Gaffar, The Small Cardinals $\mathfrak p$ and $\mathfrak u$

• Wed, May 1

• Chris Hebert, Dowker spaces

• David Webb, The Bounding and Dominating Numbers

• Jack Yoon, Martin’s Axiom

• This mathoverflow question touches on some issues about talking about infinite families of proper classes, raising much of the same issues as came up in our discussion of ZFC proving the consistency of its finite fragments. I recommend checking it out to get a better understanding of what’s going on.

• The class took a group hike on Mānana ridge trail on President’s day. A group picture can be seen here (From left to right: myself, Ikenna Nometa, Jake Fennick, Chris Hebert, Jack Yoon.)

• I strongly recommend you check out Timothy Chow’s paper “A beginner’s guide to forcing” before we start part 2. Forcing both has a lot of technical details to work through as well as being conceptually difficult to wrap one’s head around the first time one sees it. Chow’s paper is a high-level overview of forcing, and is helpful for having a big-picture idea of what’s going on before we dive into the details.