MATH 512 Advanced Topics in Logic: Infinitary Combinatorics
Spring 2022
Instructor: Dima Sinapova
Class Meets: MWF 2:00 - 2:50 in LH 101
Office: 421 SEO
Office Hours:
phone: (312)-996-2371
e-mail: sinapova@uic.edu
Description
We will cover recent developments in infinitary combinatorics, and applications of forcing techniques.
We will start by introducing combinatorial principles such as stationary reflection, Aronszajn trees, and how they interact with cardinal arithmetic
(e.g. CH, SCH). Then we will go over iterated forcing and applications to the combinatorics at $\aleph_2$.
Finally, we will cover combinatorics at successors of singulars, in particular at $\aleph_{\omega+1}$.
Prior knowledge of forcing is helpful, but nor required. I will define the necessary concepts and terminology.
Here is a tentative breakdown of topics:
- Large cardinals. Stationary reflection and the tree property.
- Cardinal arithmetic: the connection between CH, GCH, SCH and combinatorial propeties
- Forcing: basic definitions and examples including the Levy collapse, the Cohen poset, Prikry forcing. Iterated forcing
- Consistency results I: Constructions at $\aleph_2$
- Consistency results II: Constructions at $\aleph_{\omega+1}$
Homework and grading