Course Syllabus

Topic: Forcing the tree property

Course description: Forcing is a technique introduced by Cohen in the 1960s to prove the independence of the Continuum Hypothesis from the axioms of ZFC. With many further developments in the years since, it has become a central method in set theory, with wide ranging applications. Many of these applications, especially ones involving singular cardinal combinatorics, use large cardinal axioms. These are axioms of set theory that go beyond the standard system of ZFC, and typically involve elementary embeddings on the universe of sets.  

This class will explore the connections between forcing, large cardinals, and the tree property: The tree property at a cardinal \(\kappa\) states that every tree of height \(\kappa\) with levels of sizes below \(\kappa\), has a cofinal branch. It provably holds at \(\aleph_0\) and fails at \(\aleph_1\).

We will start with proofs, from large cardinals and using forcing, that the tree property can hold at \(\aleph_2\). We will continue with the much more complicated question of obtaining the tree property at successive cardinals, and the even more complicated question of obtaining the tree property on regulars in intervals that involve or overlap strong limit cardinals. These results are part of a long term quest to determine whether it is consistent that the tree property holds on all regular cardinals above \(\aleph_1\). 

Prerequisites: The class will assume knowledge of forcing, from a previous Math 223S, or from an equivalent source. The notes at http://homepages.math.uic.edu/~shac/forcing/forcing.html are a good way to become familiar with posets and generic filters. The books Set Theory, an Introduction to Independence Proofs by Kenneth Kunen, and Set Theory by Thomas Jech, provide a more comprehensive source for general forcing. 

Text: There is no required textbook. Jech's book defines many of the large cardinal axioms that we will use in class, as does the book The higher infinite by Akihiro Kanamori. Various references to research papers will be provided during the course.  

Grading: Grades will be based on two factors: attendance (roughly 30%), and presentations in class toward the end of the quarter (roughly 70%). The presentations will cover results from research papers that will be assigned in the second half of the course. Attendance is required except for medical reasons (including any covid-19 related isolation), or time zone impossibilities (meaning local times between 9pm and 8am) for zoom lectures. If this affects you please let me know ahead, and I will try to make recordings or notes available to you. During remote instruction, attendance is required with cameras turned on. The purpose of the attendance requirement is to promote a more interactive class, especially in light of remote instruction. Credit for attendance will be computed as follows: full credit for 2 or fewer missed lectures, no credit for 10 or more missed lectures, and the credit for a number between 2 and 10 interpolated linearly. 

Lecture time: The class is scheduled for MW 2-3:15pm, currently planned for Boelter 5436 except during the first two weeks which are by zoom. The zoom link is listed here (requires log in). 

Office hours: Mondays 11am-12noon, Wednesdays 10am-11am, or by appointment.