Prerequisites

The course is a combination of an introductory and an advanced course in set theory. Since we begin by developing Set Theory from its axioms the course can be taken by students without earlier experience of axiomatic set theory. We will however assume mathematical maturity, including the naïve use of sets that is very common in mathematics.

Aim of the course

To provide the students with a basic knowledge of axiomatic and combinatorial Set Theory, both in preparation of further study of the subject and to provide tools that are useful in disciplines such as General Topology, Algebra and Functional Analysis.

The course will start with an introduction to axiomatic Set Theory, based on the axioms of Zermelo and Fraenkel.

It will show how the generally well-known facts from naïve Set Theory follow from these axioms and how modern mathematics can be embedded in Set Theory.

The second part of the course will be devoted to more advanced topics in Set Theory.
Possible topics are

• The Constructible Universe
• Partition Calculus: the theorems of Ramsey, Erdös-Rado and others
• Combinatorial properties of families of subsets of the natural numbers
• Trees, stationary sets, the cub filter
• PCF theory
• Large cardinals

Lecturers

K. P. Hart, TU Delft, k.p.hart@tudelft.nl
B. Löwe, University of Amsterdam, b.loewe@uva.nl

The three-hour period will generally be divided into two hours of lecturesand a one-hour exercise class.

Homework: 30%. There will be homework assignments, weekly or bi-weekly. Homework can and should be handed in in groups of up to three students. Homework has to be handed in at the beginning of the Monday lecture (10am). Since the homework is discussed in the exercise class of the same day, late homework cannot be accepted.

Exam: 70%.

We shall use the following text:
T. Jech, Set Theory. The Third Millenium Edition

We also recommend

• K. Kunen, Set Theory. An introduction to Independence proofs.
• Y. N. Moschovakis, Notes on Set Theory
• K. J. Devlin, The Joy of Sets.

First lecture: 11 September 2017

The ZFC-Axioms, Part I.

Lecturer. K. P. Hart.

Material covered: Chapter 1, up to and including p. 11: Axiom Schema of Comprehension (false). Russell's paradox. The language of set theory. Classes. Extensionality. Pairing. The Separation Schema. Union. Power set: Cartesian products, relations, functions.

Homework set #1.

Lorenzo noticed during marking homework #1 that some students in class do not know how to write proper mathematical solutions to questions. Keep in mind that writing a mathematical argument is about convincing the reader that you understand every single step and in particular the justifications for the steps. A mathematical argument needs words, explanations and complete sentences, so that the reader understands what you are saying. In order to help you with this, Lorenzo created template solutions for homework sheet #1: do not misunderstand them as mathematical templates in the sense of the ``only correct solution'', but please understand them as writing templates as an indication what would be considered the proper level of detail to convince the reader that you understood the mathematical issues.

Second lecture: 18 September 2017

The ZFC-Axioms, Part II.

Lecturer. K. P. Hart.

Material covered: Chapter 1, pp. 12 & 13: Infinity. Construction of the natural numbers as smallest inductive set. Replacement Schema: construction of the Cartesian product without the use of the power set axiom; construction of models of Pairing without Union. Informal discussion of the Axiom of Regularity.

Homework set #2.

Solutions

Third lecture: 25 September 2017

Ordinals, Part I.

Lecturer. Benedikt Löwe.

Material covered: Properties of relations: partial orders, linear orders. Order-preserving functions and isomorphisms. Well-foundedness. Well-orders. Induction: complete induction and order induction. Basic properties of well-orders. Fundamental theorem of well-orders (without proof). Ordinals.

Fourth lecture: 2 October 2017

Ordinals, Part II.

Lecturer. Benedikt Löwe.

Material covered. Proof of the Fundamental Theorem on Wellorders. Properties of ordinals: basic properties, trichotomy, wellfoundedness of the class of ordinals, non-existence of the set of all ordinals. Transfinite Induction and Recursion. The Recursion Theorem. Applications: ordinal addition, multiplication, exponentiation. Non-commutativity of plus and times.

Fifth lecture: 9 October 2017

Ordinals, Part III & Cardinals, Part I.

Lecturer. Benedikt Löwe.

Material covered. Ordinals. Properties of ordinal operations. Cantor Normal Form: existence and uniqueness. Representation theorem for well-orders. Cardinals. Bolzano's paradoxes of the infinite. Cantor's definition of equinumerosity. Cantor's theorem. Dedekind infinity.

Homework set #5.

Sixth lecture: 16 October 2017

Cardinals, Part II.

Lecturer. Benedikt Löwe.

Material covered. Countability. The equivalence relation of equinumerosity splits the ordinals into contiguous blocks. Existence of uncountable ordinals: Hartogs's theorem.The Alephs. Operations of cardinal arithmetic. Statement of Hessenberg's theorem (no proof). Cantor-Bernstein Theorem. Cofinality, regularity and singularity. Existence of singular cardinals.

Seventh lecture: 23 October 2017

Construction of the real numbers & the Cantor-Bendixson theorem.

Lecturer. K. P. Hart.

Topics covered: Uncountablity of the real line; cardinality of the continuum: c = 2^ℵ₀; the Continuum Hypothesis; order structure of R: order-completion; Suslin's problem; topology: closed and perfect sets and the Baire category theorem.

Eighth lecture: 30 October 2017

The Axiom of Choice.

Lecturer. Benedikt Löwe.

Material covered. The Axiom of Choice and fragments of the Axiom of Choice such as the Countable Axiom of Choice. The Axiom of Dependent Choices. Applications of AC: turning surjections into injections, size of unions, regularity of successor cardinals. Zermelo's Well-Ordering Theorem and its equivalence with AC. Zorn's Lemma and its equivalence with AC.

Ninth lecture: 6 November 2017

Cardinal arithmetic.

Lecturer. K. P. Hart.

Cardinal Arithmetic and The Continuum Function up to and including 5.18.

The behaviour of the Continuum Function and how it ultimately depends on the values of the Gimel-function.

For the curious: Theorem 5.18 shows that the restrictions on the continuum function for regular cardinals that were mentioned today are the only ones. Theorem 8.12 (and its generalization in Theorem 24.1) shows that there is less freedom for singular cardinals of uncountable cofinality. Finally the last part of Chapter 24, starting at pcf theory, shows what one can say for singular cardinals of countable cofinality.

Tenth lecture: 13 November 2017

Lecturer. K. P. Hart.

Finished Cardinal Arithmetic: Exponentiation and the Singular Cardinal Hypothesis.

Stationary sets: definitions of closed unbounded and stationary sets; basic properties and Fodor's Pressing-Down Lemma (Theorem 8.7).

Eleventh lecture: 20 November 2017

Lecturer. K. P. Hart.

- Every stationary subset of a regular cardinal κ can be split into κ many stationary subsets

- Ramsey's theorem plus a look ahead at (im)possible extensions of that result

For the curious: Chapters 22 and 23 contain material related to the splitting of stationary sets; you will nedd to learn some notions from Chapter 7 (and others) to be able to read these chapters.

Twelfth lecture: 27 November 2017

Lecturer. K. P. Hart.

Finished the partition theorems: Erdös-Rado and (Erdös-)Dushnik-Miller.

For the curious: uncountable cardinals that satisfy Ramsey's Theorem are called weakly compact; see Definition 9.8, Lemmas 9.9 and 9.26, Chapter 17

Thirteenth lecture: 4 December 2017

The Axiom of Regularity & the von Neumann hierarchy, Part I.

Lecturer. Benedikt Löwe.

The Axiom of Regularity (a.k.a. Axiom of Foundation): set-version and class-version. The von Neumann hierarchy (a.k.a. "cumulative hierarchy") and its basic properties. Mirimanoff-rank and its basic properties. Proper classes and improper classes in the von Neumann universe.

Fourteenth lecture: 11 December 2017

The von Neumann hierarchy, Part II & sets of hereditary size κ.

Lecturer. Benedikt Löwe.

Material covered. Closure properties of the levels of the von Neumann hierarchy, in particular closure under Replacement. Regular strong limit cardinals. Inaccessible cardinals. Inaccessible levels of the von Neumann hierarchy are closed under Replacement. Sets of hereditarily size κ. Hereditarily finite sets are the ωth level of the von Neumann hierarchy. Closure properties of the sets H_κ. The hereditarily countable sets form a structure in which all infinite sets are of the same size.

Fifteenth lecture: 18 December 2017

Inaccessible cardinals, weakly compact cardinals, & combinatorial trees.

Lecturer. Benedikt Löwe.

Material covered. Inaccessible cardinals and H_κ. Reminder of Suslin's problem. Suslin lines and Suslin trees. Aronszajn trees. Existence of Aronszajn trees. κ-Aronszajn trees and the tree property. Weakly compact cardinals. Proof that every weakly compact cardinal has the tree property. Discussion (without proof) of large cardinal axioms and their relative strength.