Tweet
eXTReMe Tracker

Mastermath: Set Theory (spring 2013)

Exam

The exam dates are

Old exams and sample questions

Note: the exams below cover the course as taught in 2011, which put more emphasis on large cardinals (Mahlo, weakly compact) and the Axiom of Choice. Here are some comments on the last homework.

Program

Week 15: The Singular Cardinals Problem (Notes)
We do the proof of Theorem 24.18 and, if time permits, an overview of the proof of Theorem 24.33.
Week 14: The Singular Cardinals Problem (Notes)
We start to work on pcf theory; proof of Theorems 24.8. Theorem 24.8 holds without the strong limit assumption (with a much longer proof), see Theorem 2.26 in the chapter on cardinal arithmetic by Abraham and Magidor in the Handbook of Set Theory.
Here is a paper by Kojman and Shelah where the cofinal sequence of length $\aleph_{\omega+1}$ in $\prod_{n\in\omega}\aleph_n$ is used to construct a normal space whose product with the interval $[0,1]$ is not normal.
Note: if $M\prec H(\theta)$ is countable and $S\in M$ is stationary then $\delta_M\in S$ is not necessarily true: by elementarity there are stationary $S,T\in M$ that are disjoint, so $\delta_M\notin S$ or $\delta_M\notin T$. What is true is that given a stationary $S$ there are many $M$ such that $\delta_M\in S$, but this requires an extra argument.
Homework: (7.3, 7.4, 7.5, these count as one ); 24.1, 24.3, 24.6 and 24.13. Hand in in Week 15.
Week 13: Stationary sets and The Singular Cardinals Problem (Notes)
Silver's Theorem and its extension, the Galvin-Hajnal Theorem.
This book, Introduction to Cardinal Arithmetic, may be useful.
Week 12: The Axiom of Regularity and Models of Set Theory (Notes)
After a very short introduction to Model Theory we dealt with structures for the language of Set Theory. One non-standard interpretation: $\langle\mathbb{Z},{ < }\rangle$. This one satisfies the union axiom, but not the pairing axiom. The main focus was on structures where $\in$ interprets itself. We saw that $\Delta_0$-formulas are absolute for transitive sets and we used $V_\omega$, $V_{\omega+\omega}$, $V_\kappa$ ($\kappa$ inaccessible) and $H(\kappa)$ (for various $\kappa$) to see that certain axioms cannot be derived from others. The last hour was devoted to elementary substructures of $H(\kappa)$. More on elementarity can be found via this page.
Homework: 9.9 (see page 117), 9.12 (insert `almost' before `disjoint'), 12.6; Check whether the Axiom of Regularity and the Power Set Axiom hold in $\langle\mathbb{Z},{ < }\rangle$; prove the Pressing-down and $\Delta$-system lemmas for $\omega_1$ using elementary substructures of suitable $H(\kappa)$, that is: for a regressive function on $\omega_1$ and for a sequence $\langle x_\alpha:\alpha < \omega_1\rangle$ of finite subsets of $\omega_1$ respectively. Hand in in week 13 (that's May 7).
Week 11: Combinatorial set theory and The Axiom of Regularity (Notes)
An application of the $\Delta$-system lemma: the topological product $\omega^\kappa$ satisfies the countable chain condition. Almost disjoint families: three examples of almost disjoint families on $\omega$, each of cardinality $2^{\aleph_0}$. There are almost disjoint families of cardinality $\kappa^+$ on $\kappa$ (if $\kappa$ is regular) but not necessarily of cardinality $2^\kappa$. A discussion of Kurepa trees and Kurepa families and the consequences of their non-existence. Back to the Axiom of Regularity: the cumulative hierarchy $\langle V_\alpha:\alpha\in\mathrm{Ord}\rangle$ and its use in a set-theoretically sound definition of `isomorphism type'.
As to last week's homework: te hint to problem 8.5 is not quite correct. The best statement to prove by induction on $\alpha$ is: ``For every $\gamma\in\omega_1$ there is a closed (in $\omega_1$) set $A$ such that $A\subseteq S$ and the order type of $A$ is $\alpha+1$.'' In the hint, for limit $\alpha$, first take an increasing sequence $\langle\alpha_n:n < \omega\rangle$ with limit $\alpha$. Apply the inductive hypothesis to find closed sets $A(n,\xi)$ ($n\in\omega$, $\xi\in\omega_1$) such that $A(n,\xi)$ has type $\alpha_n+1$ and $\max A(n,\xi) < \min A(m,\eta)$ whenever $\xi < \eta$. Now follow the hint again.
Those who want to redo their homework may do so; hand it in in week 12 (please let me know if you intend to do so).
Week 10: Stationary sets and Combinatorial set theory (Notes, these contain an application of the $\Delta$-system lemma and some remarks about the tree property)
Every stationary subset of $\kappa$ can be split into $\kappa$ many stationary sets. Trees: Kőnig's infinity lemma and Aronszajn trees. The $\Delta$-system lemma.
Homework: 8.2, 8.5; 9.3, 9.5, 9.10. Hand in in week 11.
Note: In the hint to problem 8.5 one should find closed sets of order type $\alpha+1$, rather than of type $\alpha$.
Week 9: Combinatorial set theory and Stationary sets (Notes)
The Erdős-Dushnik-Miller theorem. Weakly compact cardinals. Closed unbounded and stationary sets. The club filter. Fodor's Lemma (a.k.a. the Pressing-Down lemma) with the Free-set lemma as an application.
Week 8: Combinatorial set theory (Notes)
Because this Tuesday is Erdős' 100th birthday we shall do some combinatorial set theory this week and deal with various extensions of the pigeon-hole principle: the partition theorms of Ramsey, and Erdős-Rado. We shall also see some limitations.
Homework: problems 5.17 (complete argument), 5.18, 5.27; 9.1 and 9.4 plus 9.13 (these count as one). Hand in in week 9
More sources:
Week 7: Axiom of Choice and Cardinal Arithmetic and The Axiom of Regularity (Notes)
The possible behaviour of the Continuum Function ($\kappa\mapsto2^\kappa$) and cardinal exponentiation ($\langle\kappa,\lambda\rangle\mapsto\kappa^\lambda$) The Axiom of Regularity and the cumulative hierarchy of sets: $\langle V_\alpha:\alpha\in\mathrm{Ord}\rangle$.
Week 6: Cardinal numbers and Cardinal Arithmetic and Axiom of Choice (Notes)
Homework: problems 3.5, 3.13; 5.3, 5.6 and 5.8. Hand in in week 7.
Fun stuff: there is a band called Axiom of Choice and the band Epoch of Unlight released an album called The Continuum Hypothesis, with a song of the same name.
Week 5: Cardinal numbers and Cardinal Arithmetic (Notes)
Week 4: Well-orderings and ordinal numbers (Notes)
Homework: problems 2.5, 2.7, 2.8, 2.11 and 2.13. Hand in in week 5.
Week 3: Well-orderings and ordinal numbers (Notes)
Introduction to well-ordered sets and ordinal numbers. Main results: any two well-ordered sets are comparable and every well-ordered set is isomorphic to an ordinal number.
Week 2: More Axioms (Notes)
The axioms of power set, infinity, replacement and regularity.
Homework: problems 1.5, 1.6, 1.7, 1.8 and 1.9. Hand in in week 3.
Note In the spirit of the book: do not assume the Axiom of Regularity when doing the exrcises.
Week 1: Axioms (Notes)
An introduction to Set Theory with an explanation of the need for axioms. The first four axioms (plus one): Extensionality, Existence of empty set, Pairing, Separation, Union.
Do try exercises 1.1 and 1.2.
Here is Bernard Bolzano's Paradoxien des Unendlichen; the bijection between the intervals $[0,5]$ and $[0,12]$ is discussed on page 29. Cantor's definition of `Menge' can be found via this link.
Here is the page for this course at mastermath.nl.

Time and Place

Starting 5 February we meet on Tuesdays from 10:15--13:00 in room S655 at the VU University, De Boelelaan 1081 (except for (calendar) week 13, when we meet in M1.43, and week 18, when there will be no class because of Koninginnedag).

Literature

We shall be using the following text: T. Jech, Set Theory. The Third Millenium Edition. Some universities have access to the on-line version of this book; check the link on the campus of your university. If it does not work directly try clicking on `Access old Springer Link' (just above the search box).

We also recommend the following books

Course in Leiden

Here is a link to a course in Set Theory in Leiden. It contains links to the original works of Cantor and others.

Other material

The webpage of the 2011 course is still available

K_dot_P_dot_Hart_at_TUDelft_dot_nl
Last modified: Friday 08-11-2013 at 09:19:43 (CET)