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

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 highlevel overview of forcing, and is helpful for having a bigpicture idea of what’s going on before we dive into the details.
Exercises

Poset combinatorics exercises. Write up your solutions and hand them in to me before we start Part 2. (Exact date pending…)
Lecture notes

Part 0 Introduction

Part ½ Logical preliminaries

Part 1.0 Inaccessible cardinals

Part 1.1 Measurable cardinals

Part 1.2 Supercompact cardinals and beyond

Part 2.1 An introduction to forcing

Part 2.2 Actually doing things with forcing

Part 3 Large cardinals and forcing
These notes are licensed under a Creative Commons AttributionNonCommercialShareAlike 4.0 International License.