Kaisa_2012_3_photo by Veikko Somerpuro
Large Cardinals 2
This is the continuation of the Large Cardinals course taught in fall 2017. Co-taught with Boban Velickovic.
Boban and JK continue the study the combinatorics of "small" large cardinals such as the weakly compact ones, their equivalent definitions in terms of trees, filters, ultraproducts, Martin's Axiom, reflection and other assorted topics. The Mycielski game and the axiom of determinacy are introduced, specifically how large cardinals (such as measurables) yield determinacy. The exercise class is co-taught with post-doc Hazel Brickhill. Forcing is not required for this course but the course Elements of Set Theory, or equivalent knowledge, is required. The course is passed by taking an exam, or with a project presented in class.
Viestit
Aikataulu
Materiaalit
Some material on Determinacy, including the Gale-Stewart Theorem, which we will cover this week.
Muu
Tehtävät
Homework set 1
Homework set 1
Homework set 2
Homework set 2
Projects
Here are some projects. You can do any of these with a partner:
1. Present the Souslin-Kleene Theorem: A set is borel iff it is analytic and co-analytic
2. Present the theorem: if there is a measurable cardinal then there are only countably many reals in L. (You should familiarise yourself with the concept of indiscernables.)
3. Present Shelah's result in the attached paper of S. Shelah called: Weakly Compact Cardinals: A Combinatorial Proof
4. (Tapio was offered and might choose this): Prove that \kappa weakly compact iff \kappa is Pi^1_1-indescribable iff [\kappa is inaccessible and L_{\kappa, \kappa} satisfies the weak compactness theorem.]
I will provide references to the standard literature if needed.
Homework set 3
Homework set 3
Solutions, homework set 2, q4&5
Solutions, homework set 2, q4&5
Homework set 4
Homework set 4
WHAT WE COVERED IN THE LECTURES
First lecture of the new quarter: We reviewed facts about measurable cardinals: e.g. that they are weakly compact. We then introduced the Axiom of Determinacy (AD), proved that AD refutes the Axiom of Choice, also that AS implies countable choice.