### Fall 2016 set theory seminar: Pcf theory

Meeting: M 12-12:50pm in SEO 427.

Pcf ("possible cofinalities") theory is a study of ultraproducts of linear orders, and is a powerful tool for proving bounds on the cardinal exponential function in ZFC; there are also applications to general topology (e.g. construction of a Dowker space) and algebra (around Whitehead's problem for free abelian groups).

We will read Abraham and Magidor's chapter "Cardinal Arithmetic" from the Handbook of Set Theory, available through springerlink.

Our goal is to reach the proof of Shelah's famous upper bound on the power of the least singular cardinal: If $\aleph_\omega$ is strong limit, then $2^{\aleph_{\omega}} < \aleph_{\omega_4}$.

Speakers:
• 8/29 - Sherwood Hachtman: Introduction, Silver's theorem.
• 9/12 - Jin Du: Posets and true cofinality; the "sandwich argument"; $(*)_\kappa$.
• 9/19 - Noah Schoem
• 9/26 - Hunter Chase
• 10/3 - Matthew Devilbliss
• 10/10 - Jin Du
• 10/17 - John Baldwin
• 10/24 - Mihail Hurmuzov
• 10/31 - Thomas Dean
• 11/7 - TBD
• 11/14 - Maxwell Levine
• 11/21 - TBD
• 11/28 - TBD