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.

Juliette Kennedy

Publicerad, 2.11.2017 kl. 17:08

Problem set 1 will be done extempore in class Nov. 3rd.

Tidsschema

I den här delen hittar du kursens tidsschema. Kontrollera eventuella andra tider i beskrivning.

DatumTidPlats
tis 31.10.2017
12:15 - 14:00
tors 2.11.2017
12:15 - 14:00
tis 7.11.2017
12:15 - 14:00
tors 9.11.2017
12:15 - 14:00
tis 14.11.2017
12:15 - 14:00
tors 16.11.2017
12:15 - 14:00
tis 21.11.2017
12:15 - 14:00
tors 23.11.2017
12:15 - 14:00
tis 28.11.2017
12:15 - 14:00
tors 30.11.2017
12:15 - 14:00
tis 5.12.2017
12:15 - 14:00
tors 7.12.2017
12:15 - 14:00
tis 12.12.2017
12:15 - 14:00
tors 14.12.2017
12:15 - 14:00

Övrig undervisning

03.11. - 10.11.2017 fre 12.15-14.00
24.11. - 15.12.2017 fre 12.15-14.00
Juliette Kennedy
Undervisningsspråk: Engelska

Material

Some material on Determinacy, including the Gale-Stewart Theorem, which we will cover this week.

Övriga

Homework set 1

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

Solutions, homework set 2, q4&5

Solutions, homework set 2, q4&5

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.