The MasterMath course Set Theory was taught remotely
during the 1st Semester 2020/21 by K P Hart and Benedikt Löwe,
assisted by
Ezra Schoen and Ned Wontner. The
course website was hosted on the
Set Theory  8EC  

1. Prerequisites. The course is a combination of an introductory and an advanced course in set theory. As a consequence, no prior knowledge of axiomatic set theory is assumed. We shall, however, assume mathematical maturity, including the naïve use of sets that is very common in mathematics. Furthermore, in this course, we shall use basic notions and results from Mathematical Logic and Model Theory and we expect students to be familiar with this material. Students who did not take an introductory course on mathematical logic can find the material in, e.g.,
2. Aim of the course. The aim is to provide the students with a basic knowledge of axiomatic and combinatorial set theory, to prepare the students for research in set theory and for using set theory as a tool in mathematical areas such as general topology, algebra and functional analysis. The course will start with a brief introduction to axiomatic set theory, the model theory of set theory (including simple independence results), and the basic theory of ordinals and cardinals. The second part of the course will be devoted to more advanced topics in set theory. This year, the focus of the advanced topics will be Large Cardinals, ranging from inaccessible via weakly compact to measurable and possibly beyond. 3. Rudiments of Axiomatic Set Theory. For UvA students, it is possible to take only the introductory component of this course if they register for the course Rudiments of Axiomatic Set Theory (UvA course code 5314RAST3Y; worth 3 ECTS credits). In that case, students follow the first seven lectures and do the corresponding course work and sit an exam on 18 December 2020. It is not yet known what the form of this exam is going to be. Only one of the two courses (Set Theory and Rudiments) can be registered, so students registered for both need to decide by early December 2020 whether to sit the Rudiments or the Set Theory exam. Even if you plan to sit the Rudiments exam, students are encouraged to follow lectures 8 to 15 of Set Theory, both in order to deepen their knowledge in set theory and to keep the option of getting credit for the full course 4. Lecturers & Teaching Assistants. Benedikt Löwe will be lecturing the first seven lectures, covering the basic material; after that, KP Hart will take over as the lecturer. The Teaching Assistants are Ezra Schoen and Ned Wontner. 5. Assessment policy. The final course grade consists of 10% course work and 90% exam, rounded to full numbers. The precise form of the exam is still unknown due to the current pandemic situation; we hope to know more in October. 5.1. Resits. Final grades will be computed separately for the first exam (.1 x homework + .9 x exam) and the resit (.1 x homework + .9 x resit) and then submitted separately to the local administrations. The local administrations then apply the resit grading rules relevant for the individual student: in some programmes, this will be the maximum of the two grades, in others it will be the later of the two grades, even if that is lower than the earlier one. Students are advised to check their own institution's rules before deciding whether to take the resit exam or not. 5.2. Course work. Course work consists of two components: home work and group interactions. The course work score will be the average of the scores for these two components. 5.2a. Home work. Each week, we'll assign one quiz with multiple choice questions to be answered directly on the course website and additional questions for which solutions need to be submitted as pdf file via the course website. For the additional questions, collaboration between students is encouraged, but each student should write and submit their solutions independently. The answers to both the MC questions and the additional questions are due at 2pm on the next Monday, i.e., just before class starts. Each correct answer to a multiple choice question gives you one point; incorrect answers give no points. Each submitted answer to an additional question gives one point, independent of whether it is correct or not. The score for the homework will be based on the percentage of total points scored. 5.2b. Group interaction. Each week, students will be assigned to small collaborative groups in which they will work on one particular question in the presence of a teaching assistant (via Zoom). All students are expected to participate in these group interactions every week. You sign up for these Group Interactions via this webpage. The score for group interactions is the number of these sessions that you actively participated in (with a maximum of 10). [Students who do the Rudiments of Axiomatic Set Theory exam and thus only participate in seven of these group interactions get their score for this component doubled, again, with a maximum of 10.] 6. Literature. We are going to follow the monograph by Thomas Jech: Set Theory.  
Recordings. 
Recordings of Lectures #8 to #15 can be found on vimeo. The password is 5RyQ. 
Exam. 
The written exam for the course Set Theory will take place on Monday 25 January 2021 from 2pm to 5pm. The resit exam will be on Monday 1 March 2021 from 2pm to 5pm. Both the exam and the resit will be online open book exams. This means that you are allowed and encouraged to use your notes and books while sitting the exam. However, you are not allowed to communicate with any other people by any other means during the exam. Please read the following practical and structural information about the exam(s) carefully: PRACTICAL MATTERS.
EXAM STRUCTURE.
You can find a Template Exam and a comment on the Template Exam here: Template Exam & Comment. The template has exactly the same structure as the exam and the resit and gives an indication of what the exams will look like. The comment gives answers to the questions and some information about how the answers will be marked. 
First lecture. 7 September 2020. 
First lecture. Lecturer: Benedikt Löwe. Motivation: set theory as a language of mathematics, set theory as a foundations of mathematics, set theory as a subfield of mathematics. The two protagonists of set theory: cardinals and ordinals (brief historical motivation). The axiomatic method (comparison between geometry and set theory). Attempts of definitions of the concept of set. The formal language of set theory and its graph models. Axiom of Extensionality (contrast with ordered tuples and multisets). Axioms of Pairing, Union, and Power Set. Checking the meaning of these axioms in simple graph models. Language expansions by relation and function symbols. Standard abbreviations in set theory. The Axiom Scheme of Comprehension. Russell's Paradox. The Axiom Scheme of Separation. Lecture Notes. Group Interaction #1. The pdf file with the instructions for Group Interaction #1 to be held in the first week of classes is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Please select your time slots using the link Selection of time slots below. Quiz #1. You will find three multiple choice questions below under the heading Quiz #1. Please answer these questions before Monday, 14 September 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #1. You will find three homework questions below under the heading Homework Assignment #1. Please submit your answers to these questions as a single pdf file before Monday, 14 September 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Second lecture. 14 September 2010. 
Second lecture. Lecturer: Benedikt Löwe. FST: Finite Set Theory. FST implies the existence of a unique empty set. FST implies that there is no set of all sets. Discussion of the existence of a locally finite model of FST (without proof; cf. Group Interaction #1). Reconstruction of abstract mathematics in FST: ordered pairs (Kuratowski), relations, functions, injectivity, surjectivity, equivalence relations, quotients. Models of FST cannot be finite: the successor operation, the notion of a transitive set, proofs by induction and constructions by recursion in informal mathematics. Inductive sets. The Axiom of Infinity. Zermelo set theory Z. Existence of a least inductive set \(\mathbb{N}\). Properties of the least inductive set: \(\in\) is a total order on \(\mathbb{N}\) (cf. Homework Assignment #2). The Recursion Theorem on \(\mathbb{N}\) and its proof (via germs; cf. Group Interaction #2). Lecture Notes. Group Interaction #2. The pdf file with the instructions for Group Interaction #2 to be held in the second week of classes is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Please select your time slots using the link Selection of time slots below. Quiz #2. You will find three multiple choice questions below under the heading Quiz #2. Please answer these questions before Monday, 21 September 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #2. You will find three homework questions below under the heading Homework Assignment #2. Please submit your answers to these questions as a single pdf file before Monday, 21 September 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Third lecture. 21 September 2020. 
Third lecture. Lecturer: Benedikt Löwe. Zermelo set theory Z captures most of mathematical practice (abstract maths and the natural numbers \(\mathbb{N}\) together give us the integers, the rationals, and the reals as discussed in Group Interaction #3). The exception are recursive definitions where the range is not determined in advance. The Axiom Scheme of Replacement: version for partial functional formulas (as in Jech) and version for total functional formulas, and their equivalence (in Z). Proof of the Recursion Theorem without range fixed in advance (using the Axiom Scheme of Replacement). The Axiom of Regularity (or Foundation). Regularity implies that there are no sets that contain themselves as elements. Partial and linear orderings (strict and nonstrict) and their basic notions. The least number principle for the natural numbers. Wellorderings. Examples and nonexamples. Order sums and products and the fact they preserve wellorders. Basic properties of functions on wellorderings (Lemma 2.4, Corollaries 2.5, 2.6, & Lemma 2.7 in Jech). Induction and recursion on wellorders. Fundamental Theorem for Wellorders (Theorem 2.8 in Jech) and a proof via the Recursion Theorem. Lecture Notes. Group Interaction #3. The pdf file with the instructions for Group Interaction #3 to be held in the second week of classes is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Please select your time slots using the link Selection of time slots below. Quiz #3. You will find three multiple choice questions below under the heading Quiz #3. Please answer these questions before Monday, 28 September 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #3. You will find three homework questions below under the heading Homework Assignment #3. Please submit your answers to these questions as a single pdf file before Monday, 28 September 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Fourth lecture. 28 September 2020. 
Fourth lecture. Lecturer: Benedikt Löwe. The ordering of wellorders: every nonempty set of wellorders has a minimal element. Ordinals: basic properties and examples: natural numbers and successor operation. Linear ordering of the ordinals by the subset relation. Unions of sets of ordinals are ordinals. There is no set of all ordinals. Successor ordinals and limit ordinals. Hartogs's Theorem and the Hartogs aleph. Representation theorem for well orders (every wellorder is isomorphic to a unique ordinal). Transfinite induction and recursion on the ordinals. Ordinal arithmetic: addition, multiplication, and exponentiation. Asymmetry of the recursion definitions. Examples: \(\omega+1 > 1+\omega = \omega\) and \(\omega\cdot 2 = \omega + \omega > 2\cdot\omega = \omega\). Lecture Notes. Group Interaction #4. The pdf file with the instructions for Group Interaction #4 is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Please select your time slots using the link Selection of time slots below. Quiz #4. You will find three multiple choice questions below under the heading Quiz #4. Please answer these questions before Monday, 5 October 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #4. You will find four homework questions below under the heading Homework Assignment #4. Please submit your answers to these questions as a single pdf file before Monday, 5 Octobber 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Fifth lecture. 5 October 2020. 
Fifth lecture. Lecturer: Benedikt Löwe. More on ordinals: applications of Cantor Normal Form (cf. Group Interaction #4) for calculations with ordinals; synthetic definitions of ordinal addition and multiplication (proof of the equivalence for addition; for ordinal exponentiation, cf. Homework Sheet #5); normal operation and existence of fixed points. Cardinality. CantorSchröderBernstein Theorem (for a proof, cf. Homework Sheet #5). Finite cardinals, Dedekindfiniteness. Wellorderability. Characterisation of wellorderability. Some cardinality calculations: countability of \(\mathbb{Z}\) and \(\mathbb{Q}\); Cantor's Theorem; the power set and characteristic functions; cardinality of \(\mathbb{R}\). Cardinalities for wellorderable sets and definition of cardinal numbers. The aleph sequence: proof that the sequence contains all infinite cardinals. Lecture Notes. Group Interaction #5. The pdf file with the instructions for Group Interaction #5 is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #5. You will find three multiple choice questions below under the heading Quiz #5. Please answer these questions before Monday, 12 October 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #5. You will find three homework questions below under the heading Homework Assignment #5. Please submit your answers to these questions as a single pdf file before Monday, 12 October 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Sixth lecture. 12 October 2020. 
Sixth lecture. Lecturer: Benedikt Löwe. Recapitulation of the aleph hierarchy. Notation: \(\aleph_\alpha\) vs. \(\omega_\alpha\). Existence of aleph fixed points and the fact that the first aleph fixed point is a countable union of smaller cardinals. The Axiom of Choice. Uses in mainstream mathematics: countable unions of countable sets are countable. Situations in which choice functions exist without a use of the AC. Remarks about consequences of AC in algebra and analysis. The product notation for the set of choice functions. Zermelo's Wellordering Theorem and its proof in ZFC. The Wellordering Theorem implies AC. The Comparison Principle CP and its equivalence to AC. Successor and limit cardinals. Cofinality and basic properties of the cofinality function. Regular and singular cardinals. ZFC proves that every successor cardinal is regular. Cardinal arithmetic: addition and multiplication trivialise (due to Hessenberg's theorem; cf. Group Interaction #5). Exponentiation: definition and consequences of Cantor's theorem. Lecture Notes. Group Interaction #6. The pdf file with the instructions for Group Interaction #6 is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #6. You will find three multiple choice questions below under the heading Quiz #6. Please answer these questions before Monday, 19 October 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #6. You will find three homework questions below under the heading Homework Assignment #6. Please submit your answers to these questions as a single pdf file before Monday, 19 October 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Seventh lecture. 19 October 2020. 
Seventh lecture. Lecturer: Benedikt Löwe. More on cofinalities: calculating cofinalities of concrete limit cardinals. The question of regular limit cardinals: weakly inaccessible and strong inaccessibles. Weakly inaccessibles are aleph fixed points. The von Neumann hierarchy and its basic properties. Graphical representation of the set theoretic universe. Theorem and Principle of \(\in\)induction. Proof that the von Neumann hierarchy exhausts the universe. Classes: represented classes and proper classes. Proof that a definable class is represented if and only if it is bounded in the von Neumann hierarchy. Model theoretic properties of von Neumann ranks: all limit ranks are models of FST; all limit levels \(>\omega\) are models of Z. Discussion of the Replacement axiom: proof that for countable \(\alpha\), \(\mathbf{V}_\alpha\) cannot be a model of the Replacement axiom scheme. Theorem of Shephardson (without proof). Lecture Notes. Group Interaction #7. The pdf file with the instructions for Group Interaction #7 is here: pdf file. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #7. You will find three multiple choice questions below under the heading Quiz #6. Please answer these questions before Monday, 26 October 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #7. You will find four homework questions below under the heading Homework Assignment #7. Please submit your answers to these questions as a single pdf file before Monday, 26 October 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Eighth lecture. 26 October 2020. 
Eighth lecture. Lecturer: K.P. Hart. A characterization of cofinality. Definition of sum, product and exponentiation of two cardinals; also sums and products of arbitrary families of cardinals. Statement of the basic problem: how does the \(\gamma\) in \(\aleph_\alpha^{\aleph_\beta}=\aleph_\gamma\) depend on \(\alpha\) and \(\beta\)? There is considerable freedom (or uncertainty?), but there are some restrictions on the values of the Continuum Function and Exponentiation. We find the restrictions and freedoms that can be established by elementary means. Lecture Notes. Group Interaction #8. The pdf file with the instructions for Group Interaction #8 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #8. You will find three multiple choice questions below under the heading Quiz #8. Please answer these questions before Monday, 2 November 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #8. You will find four homework questions below under the heading Homework Assignment #8. Please submit your answers to these questions as a single pdf file before Monday, 2 November 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Ninth lecture. 2 November 2020. 
Ninth lecture. Lecturer: KP Hart. Filters (and their dual ideals) and ultrafilters. Definitions and basic properties. Existence of ultrafilters. Ultrafilters on \(\omega\): Ramsey ultrafilters. \(\kappa\)completeness of filters: the question about the existence og \(\sigma\)complete ultrafilters foreshadows measurable cardinals. The filter generated by the closed unbounded sets on a regular cardinal \(\kappa\): it is \(\kappa\)complete and closed under diagonal intersections. Stationary sets and Fodor's PressingDown Lemma. Lecture Notes. Group Interaction #9. The pdf file with the instructions for Group Interaction #9 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #9. You will find three multiple choice questions below under the heading Quiz #9. Please answer these questions before Monday, 11 November 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #9. You will find four homework questions below under the heading Homework Assignment #9. Please submit your answers to these questions as a single pdf file before Monday, 11 November 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Tenth lecture. 9 November 2020. 
Tenth Lecture. Lecturer: K. P. Hart. Combinatorial Set Theory: Ramsey's Theorem, the ErdösRado theorem and the Ramsey property of selective ultrafilters. Lecture Notes. Group Interaction #10. The pdf file with the instructions for Group Interaction #10 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #10. You will find three multiple choice questions below under the heading Quiz #10. Please answer these questions before Monday, 16 November 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #10. You will find four homework questions below under the heading Homework Assignment #10. Please submit your answers to these questions as a single pdf file before Monday, 16 November 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Eleventh lecture. 16 November 2020. 
Eleventh Lecture. Lecturer: K. P. Hart. More Combinatorial Set Theory: the ErdösDushnikMiller theorem, König's Infinity Lemma, Aronszajn trees, the tree property and weakly compact cardinals. Lecture Notes. Group Interaction #11. The pdf file with the instructions for Group Interaction #11 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #11. You will find three multiple choice questions below under the heading Quiz #10. Please answer these questions before Monday, 23 November 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #11. You will find four homework questions below under the heading Homework Assignment #11. Please submit your answers to these questions as a single pdf file before Monday, 23 November 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Twelfth lecture. 23 November 2020. 
Twelfth Lecture. Lecturer: K. P. Hart. What are Large Cardinals anyway? Weak Compactness and the tree property. Elementary properties of measurable cardinals: inaccessibility, weak compactness, and normal measures. Lecture Notes. RecommendationThis would be a good time to refresh your knowledge of Model Theory, in particular the notions of elementary substructures and ultrapowers. Group Interaction #12. The pdf file with the instructions for Group Interaction #12 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #12. You will find three multiple choice questions below under the heading Quiz #12. Please answer these questions before Monday, 30 November 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #12. You will find four homework questions below under the heading Homework Assignment #12. Please submit your answers to these questions as a single pdf file before Monday, 30 November 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Thirteenth lecture. 30 November 2020. 
Thirteenth Lecture. Lecturer: K. P. Hart. Ultrapowers of the Universe Definition of ultrapowers; wellfoundedness from \(\sigma\)complete ultrafilters. The Mostowski collapse of the ultrapower by a \(\kappa\)complete ultrafilter on \(\kappa\) and a few useful properties. Proof using ultrapowers that there are stationarily many weakly compact cardinals below a measurable cardinal. and applications. Lecture Notes. Group Interaction #13. The pdf file with the instructions for Group Interaction #13 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #13. You will find three multiple choice questions below under the heading Quiz #13. Please answer these questions before Monday, 7 December 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #13. You will find four homework questions below under the heading Homework Assignment #13. Please submit your answers to these questions as a single pdf file before Monday, 7 December 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Fourteenth lecture. 7 December 2020. 
Fourteenth Lecture. Lecturer: K. P. Hart. Why weakly compact?. Finishing up on Ultrapowers of the Universe. An explanation of the name weakly compact cardinal: it is related to a version of the Compactness Theorem for large languages, specifically languages of type \(\mathcal{L}_{\kappa,\omega}\) and \(\mathcal{L}_{\kappa,\kappa}\). Lecture Notes. Group Interaction #14. The pdf file with the instructions for Group Interaction #14 is here. Note that you are not expected to prepare for the group interaction sessions, so there is no need to look at the pdf file before starting the group interaction sessions. Quiz #14. You will find three multiple choice questions below under the heading Quiz #14. Please answer these questions before Monday, 14 December 2020, 2pm. You get one point per correct answer and no points for incorrect answers. jpg file. Homework Assignment #14. You will find four homework questions below under the heading Homework Assignment #14. Please submit your answers to these questions as a single pdf file before Monday, 14 December 2020, 2pm. You get one point per attempted question, independent on whether your answer is correct or not. pdf file. 
Fifteenth lecture. 14 December 2020. 
Fifteenth Lecture. Lecturer: K. P. Hart. Indescribable! We finish up the Weak Compactness Theorem and move on to (in)describability: Weakly compact cardinals are \(\Pi^1_1\)indescribable. We'll see what that means and how we can use it. Lecture Notes. Recommended exercises: from Chapter 17 the following exercises are useful.
