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

Julkaistu, 2.11.2017 klo 17:08

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

### Aikataulu

Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

PäivämääräAikaOpetuspaikka
Ti 31.10.2017
12:15 - 14:00
To 2.11.2017
12:15 - 14:00
Ti 7.11.2017
12:15 - 14:00
To 9.11.2017
12:15 - 14:00
Ti 14.11.2017
12:15 - 14:00
To 16.11.2017
12:15 - 14:00
Ti 21.11.2017
12:15 - 14:00
To 23.11.2017
12:15 - 14:00
Ti 28.11.2017
12:15 - 14:00
To 30.11.2017
12:15 - 14:00
Ti 5.12.2017
12:15 - 14:00
To 7.12.2017
12:15 - 14:00
Ti 12.12.2017
12:15 - 14:00
To 14.12.2017
12:15 - 14:00

### Muu opetus

03.11. - 10.11.2017 Pe 12.15-14.00
24.11. - 15.12.2017 Pe 12.15-14.00
Juliette Kennedy
Opetuskieli: englanti

### Materiaalit

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

## Muu

### Tehtävät

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.