### Viestit

### Aikataulu

### Materiaalit

Enderton's text called Elements of Set Theory, plus Kunen's Set Theory: An Introduction to Independence Proofs are the main text of the course. However not all subject matter is covered in these texts.

SPECIALIZED COURSE MATERIAL WILL BE POSTED AS THE COURSE PROGRESSES.

## Muu

### Tehtävät

#### Example class Sept. 8. 2 questions:

1. Prove that is kappa and lambda are 2 infinite cardinals, kappa x lambda = kappa + lambda = max(kappa, lambda)

2. Construct the V_alpha's "piecewise", by transfinite recursion on a large enough delta. Then contract them in one go, all at once.

#### TOMORROW SEPT: 8th example class

This will be an ordinary lecture. And a very important one! As tomorrow we start using Kunen's book Axiomatic Set Theory.

#### SEPTEMBER 15 homework:

1. Prove, in Kunen's theorem 9.3 (transfinite recursion):

A. Uniqueness: if G_1 and G_2 both satisfy: for all alpha, [ G (alpha) = F ( G restricted to alpha ) ] , then for all alpha, G_1(alpha)=G_2(alpha).

B. As in the uniqueness proof, if g is a delta-approximation and g’ is a delta'-approximation, then g and g' agree on the restriction of their domain to delta intersect delta'.

2. Construct ordinal exponentiation by transfinite recursion on Ord. What is omega^(omega ^ omega)?

#### September 22 homework

Prove theorem 9J on page 254, Enderton. On Enderton page 256, 257, do: 5,6,9,11

#### October 6 homework.

#### Delta lemma; homework set 4

#### Homework set 5 DUE TUESDAY OCTOBER 10

### WHAT WE COVERED IN THE LECTURES

SEPTEMBER 5: We refreshed our knowledge of the basics of the theory of cardinals and ordinals, up through the construction of the V_alpha's. The aim in the near future is to finish Enderton's book, which we will do in the next 3 lectures, approximately.

SEPTEMBER 8: We covered transfinite INDUCTION and transfinite RECURSION theorems in Kunen, i.e. theorems 9.2. and 9.3 on pages 24,25. We also defined ordinal addition by trans. recursion.

September 12th: We proved that every cardinal number is an Aleph_alpha, by transfinite induction. We then identified, for a ZFC axiom phi, for which alpha does V_alpha satisfy phi. All axioms were treated except replacement.

Sept. 14: We showed that V_kappa, for kappa inaccessible, is a model of all the ZFC axioms. Introduced the concept of cofinality.

Sept. 19: More on cofinality! We proved: cof(kappa) is always a regular cardinal; if lambda is a limit ordinal then cof(lambda)=cof(Aleph_lambda); König's theorem.

Sept. 21: We proved König's theorem again, clarifying the "sticky point". (By the way, did we use the axiom of choice in the proof?) We did some cardinal arithmetic, using cofinality. (See homework set 3 for the proof.)

Sept. 28 We defined "club subset of \kappa". We proved that the intersection of 2 club sets is club; we proved that the intersection of fewer than \kappa club sets is club. We did another cardinal exponentiation calculation, under certain cofnality assumptions; we (informally) discussed some definitions of large cardinals Mahlo, weakly inaccessible, measurable. For the material on club and stationary sets, we rely on Jech's book Set Theory (3rd ed.), chapter 8.

Sept. 29th: After review of the basics of clubs, we proved that the diagonal intersection of club sets is a club.

October 3,5: We introduce the concepts of Mahlo and weakly compact cardinals. We proved Fodor's lemma, and then we proved the delta lemma using Fodor's lemma; we showed/will show that weakly compact cardinals are inaccessible and have the tree property; if a cardinal is inaccessible and has the tree property, then it is weakly compact. We proved that \omega has the tree property.

Last lectures of the semester: Existence of an Aronszajn tree on \omega_1; Background on the measure problem;

Definition of two valued measure and ultrafilter, showing they are essentially the same; Definition of measurable cardinal; Proof that a measurable cardinal is regular, inaccessible, weakly compact; Definition and brief discussion of L;

Proof of Mostowski collapse; Definition of ultrapower of V; Los theorem; Proof that if there is measurable cardinal then there is an elementary embedding from V into an inner model with critical point the measurable; Scott's theorem

### Kuvaus

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Set Theory

Master studies

Basic knowledge of the theory of large cardinals, their equivalents in terms of tree properties, compactness etc

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Large cardinals, trees, compactness properties

Lecture notes

Lectures

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later