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 expect students to be familiar with this material. Knowledge of very basic model theory, i.e., compactness arguments, LöwenheimSkolem theorems, etc. (as usually taught in an introductory course on mathematical logic) will be indispensible. The following is a list of textbook accounts of the basics of mathematical logic:
We should like to remind students with a nonmathematical background that this is a mathematics course at the Master's level and is primarily aimed at students with an undergraduate degree in mathematics. Since very little concrete mathematical knowledge is required (beyond what is outlined above), we believe that the course is accessible to students with a nonmathematical background, provided that they have the required mathematical maturity. 2. Aim of the course. The aim is to provide the students with a basic knowledge of axiomatic set theory, combinatorial set theory, and model constructions in 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 the method of Forcing. We intend to show how this method may be used to prove the consistency and independence of various settheoretical statements such as the Continuum Hypothesis. 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 sit the exam on 20
December 2021. Only one of the two courses (Set Theory and Rudiments) can be taken for credit, so students registered for both need to decide by late November or early December 2021 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 Steef Hegeman and Robert Paßmann. 5. Assessment policy. The final course grade consists of 10% course work and 90% exam, rounded to full numbers. 5.1.
Exams. All exams relating to this course (including the Rudiments exams)
will be open book, i.e., you are allowed to use your notes and your books during
the exam. They are currently all scheduled to be in person exams. Whether the pandemic
situation will allow that or not at the time of the exam, remains to be seen. If necessary,
the exams will switch to an online exam, but not change the exam structure and content. 5.2. Resits. Final grades will be computed separately for the first exam (.1 x course work + .9 x exam) and the resit (.1 x course work + .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. 5.3. Course work. Course work consists of three components: (a) regular home work, (b) understanding and explanation tasks, and (c) group interactions. The course work score will be the average of the scores for these three components. 5.3a.
Regular home work (H). Regular home work consists of fourteen homework sets
(H1 to H14) with three to five questions per week. Each submitted answer
to an additional question gives one point, independent of whether it is correct or not. Home
work is due at 2pm on Monday one week after it is posted (at the beginning of the lecture).
Home work questions are not marked, but the teaching assistants will identify good solutions
which will be posted as examples of good solutions (with the explicit permission of the
authors). For Rudiments, only the first seven homework sheets (H1
to H7) count. 5.2b. Understanding & Explanation tasks (U). Every two weeks, we shall have one question that asks students to explain in their own words a phenomenon encountered in lectures. In total, there will be six U sheets (U1 to U6). More details will be given on the sheet U1. Each of the Understanding & Explanation tasks will be marked out of 3½ points where a satisfactory answer (no major errors, all relevant points mentioned) will get 3 points and particularly good answers get full points. For Rudiments, only the first three sheets (U1 to U3) count. 5.2c.
Group interactions (G). 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). There will be a total of thirteen group interaction sheets
(G1 to G13). 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). For Rudiments, the number is
doubled (still with a maximum of 10). 6. Literature. We are going to follow the monograph by Thomas Jech: Set Theory. PDF files of chapters from Jech's book will be provided on this page. For the last section on Forcing, we shall be using Kenneth Kunen, Set Theory. An introduction to Independence proofs. Studies in Logic and the Foundations of Mathematics, 102. NorthHolland. 1980. 

Schedule 
The course Set Theory will be taught fully online. Each week, we shall give the Zoom link here on the page. The lectures will be recorded and uploaded via vimeo, but the upload may take a few days. 
Lecture Videos 
The lecture videos will be provided on panopto. In order to get access to the lecture videos, please follow these steps: STEP 1. Create a free panopto account. For this, go to and either sign up with any google / gmail account you have and wish to use for this or sign up with any other email address. (If you do the latter, you'll need to confirm your email address later to activate your account.) STEP 2. Give us the email address used for your panopto account. Just send an email to loewe@math.unihamburg.de giving your name and your email address. We will then send you an invitation to watch an brief video. STEP 3. Receive the invitation by email. The invitation will look roughly like this. Click on play to watch the video. STEP 4. Watch the video. The video contains a description of the remaining steps. STEP 5. You now need to make sure that we are connected. This is necessary to make sure that we can share the lecture videos with you. Log on to your panopto account and go to the management of your account. Click on your name in the upper right corner. STEP 6. A little popup appears that gives you the option of signing out and managing the user settings. Choose the latter. STEP 7. In the user settings, you see a menu on the left hand side. Choose connections. STEP 8. Under Manage Connections, you can either see that you are already connected to the account dbl25@cam.ac.uk or you have an outstanding invitation from that account. In the first case, all's fine; in the second case, please accept that invitation. STEP 9. Now you have to wait. Once we are connected, we can share the video folder with you, but this needs to be done manually, so you'll have to wait until we have done this. STEP 10. Once that has happened, all new videos should appear in your panopto account to view. In case of any questions, please send an email. 
First lecture: 13 September 2021 
Lecturer: Benedikt
Löwe.
First Lecture. Administrative matters: Literature (Jech & Kunen). The exam. The course work component (homework, understanding tasks, group interactions). Overview of the course. Major axiom systems. The two protagonists of set theory: ordinals and cardinals. The axiomatic method. Euclid. The parallel postulate. Hilbert's "chairs, tables, and beermugs". Affine planes and their five axioms. Proof sketch that the first four axioms do not imply the fifth. Axioms of Set Theory. The language of set theory LST. LST structures: directed graphs. The empty set axiom and the axiom of extensionality. Validity of these axioms in all one and two element directed graphs. The axioms of Pairing and Union. No finite graph can be a model of Ext, Emp, and Pair. The Comprehension Axiom Scheme. Proof of Russell's Theorem that no graph can be a model of the Comprehension Axiom Scheme. The Separation Axiom Scheme. Preview: Separation implies that there is no set of all sets. Lecture Notes. Homework Sheet #1: pdf file. 
Second lecture: 20 September 2021 
Lecturer: Benedikt
Löwe.
Second Lecture. Separation implies Empty Set. Axioms are infinite collections of axioms: in general, no claim that they cannot be reduced to a single axiom (finite expressibility). Separation implies that there is no universal set. Application: no set of all singletons. Subsets and the power set, the power set axiom. The axiom system FST. Language extensions by definable function symbols and constant symbols in FST. Locally finite graphs: there is a locally finite model of FST (no proof, cf. Homework Sheet #2). Reconstruction of abstract mathematics in FST: the Kuratowski pair, the Cartesian product, relations and their properties, functions, structures, equivalence relations, quotients. Infinity: Dedekindinfinity, inductive sets, the axiom of infinity, Zermelo set theory Z. Definition of the natural numbers, the principle of induction, transitive sets. Basic properties of natural numbers. Lecture Notes. Homework Sheet #2. pdf file. Understanding & Explanation Sheet #1. pdf file. Group Interaction Sheet #1. pdf file. 
Third lecture: 27 September 2021 
Lecturer: Benedikt Löwe.
Third Lecture. Notation for individual natural numbers: 0, 1, 2, 3. The order relation on the natural numbers. Further properties of the natural numbers. Finiteness and infinity: proof that the set of natural numbers is infinite. Arithmetic on the natural numbers: synthetic and inductive definitions of addition. Disjoint unions. The Recursion Theorem. Synthetic and inductive definitions agree. The Least Number Principle. Wellfoundedness. The natural numbers satisfy the least number principle. Order inductive sets. The principle of order induction. Reconstruction of ordinary mathematics: the integers. Noncanonicity: the integers are only defined up to isomorphism. Countability. Proof that every countable and infinite set is countably infinite. Ordertheoretic recursion theorem (without proof). Any countably infinite set can be made isomorphic to the integers. The rationals and the reals (cf. Group Interaction #2). Other sets that satisfy the principle of order induction: the successor of the set of natural numbers. Lecture Notes. Homework Sheet #3. pdf file. Group Interaction Sheet #2. pdf file. 
Fourth lecture: 4 October 2021 
Lecturer: Benedikt Löwe.
Fourth Lecture. Wellorders. Examples. Initial segments, proper initial segments. Properties of isomorphisms of wellorders: rigidity, ordering of wellorders. Fundamental Theorem on Wellorders. Ordinals. Examples and properties. Ordering of ordinals. Wellfoundedness of ordinals. There is no set of all ordinals. Trying to get the second infinite limit ordinal: our recursion theorem is not good enough. Axiom of Replacement. The axiom system ZF_{0}. Recursion Theorem without fixed range. Proof of the existence of ω+ω. Successor ordinals and limit ordinals. Transfinite induction and recursion on a fixed ordinal. Notation for transfinite induction and recursion on the ordinals. Ordinal addition, multiplication, and exponentiation and some properties. Lecture Notes. Homework Sheet #4. pdf file. Understanding & Explanation Sheet #2. pdf file. Group Interaction Sheet #3. pdf file. 
Fifth lecture: 11 October 2021 
Lecturer: Benedikt Löwe.
Fifth Lecture. Ordinal arithmetic: swallowing from the left, distributivity phenomena. Ordinal operations. Normal ordinal operations and fixed points. Fixed points of addition and multiplication (gamma and delta numbers). Representation Theorem for wellorders. Hartogs's Theorem. Coding of ordinals injecting into X by subsets of X^{2}. HartogsAlephs. The first uncountable ordinal. Sets that contain themselves as element. The von Neumann hierarchy or cumulative hierarchy: basic properties. Graphical representations of the von Neumann hierarchy. Mirimanoff rank. The Axiom of Foundation (or Regularity). The axiom system ZF. The principle of epsiloninduction. Transitive closures. Foundation implies epsiloninduction. Foundation implies that every set is in the cumulative hierarchy. Consistency proofs using the cumulative hierarchy: validity of FST in limit levels of the von Neumann hierarchy; validity of Z in levels of the von Neumann hierarchy. The smallest von Neumann model of Z does not satisfy Replacement. Lecture Notes. Homework Sheet #5. pdf file. Group Interaction Sheet #4. pdf file. 
Sixth lecture: 18 October 2021 
Lecturer: Benedikt Löwe.
Sixth Lecture. Gödel's Incompleteness Theorem and the set theory in von
Neumann ranks. Invalidity of the Representation Theorem for Wellorders in
V_{ω+ω}. Replacement in V_{ω1}. The Alephs, initial ordinals,
cardinals. The alephs are exactly the infinite cardinals. Successor cardinals, limit
cardinals. Equinumerosity. The CantorSchröderBernstein Theorem (no proof).
Wellorderability and cardinals as canonical representatives for equinumerosity classes.
History of Zermelo's Wellordering Theorem. Choice functions. The Axiom of Choice. Zermelo's
Wellordering Theorem. Proof of AC from Zermelo's Wellordering Theorem. Equivalents
and consequences of the Axiom of Choice: comparability, countable unions of countable sets
are countable, every vector space has a basis, the BanachTarski paradox. Regularity and
singularity. All successor cardinals are regular. Lecture
Notes. Homework Sheet #6. pdf file. Understanding & Explanation Sheet #3. pdf file. Group Interaction Sheet #5. pdf file. 
Seventh lecture: 25 October 2021 
Lecturer: Benedikt Löwe.
Seventh Lecture. Sizes of sets without \(\mathsf{AC}\). Scott's Trick. Cardinalities. Examples of singular limit cardinals; proof that regular limit cardinals are aleph fixed points. Existence of aleph fixed point; the smallest aleph fixed point is singular. Weakly inaccessible cardinals. Cofinality. Cardinality arithmetic: addition and multiplication. Cardinal exponentiation and its divergence from ordinal exponentiation. Cantor's theorem. \(\lambda^{\kappa} = 2^\kappa\) for all \(\lambda\leq\kappa^+\). The Continuum Problem: \(\mathsf{CH}\) and \(\mathsf{GCH}\). Independence of \(\mathsf{CH}\) (without proof). Strongly inaccessible cardinals. If \(\kappa\) is strongly inaccessible, the \(\mathbf{V}_\kappa\models\mathsf{ZF}\). If \(\kappa\) is strongly inaccessible, the \(\mathbf{V}_\kappa\) satisfies second order replacement. \(\mathsf{ZFC}\) cannot prove that there are strongly inaccessible cardinals. Lecture Notes. Homework Sheet #7. pdf file. 
Eighth Lecture: 1 November 2021 
Lecturer: K. P. Hart. Eighth Lecture: Cardinal Arithmetic. What can be proved about cardinal exponentiation by relatively elementary means. Arbitrary sums and products of cardinals and their use in the computation of the Continuum Function and general powers. How these computations proceed and how the Gimel function appears. Lecture Notes: before the lecture; during the lecture. Homework Sheet #8. PDF Group Interaction Sheet #7. PDF Understanding & Explanation Sheet #4. PDF; the same, but with a solution. 
Ninth Lecture: 8 November 2021 
Lecturer: K. P. Hart. Ninth Lecture. We shall deal with closed unbounded (cub) and stationary subsets of regular uncountable cardinals. A major result will be Fodor's Pressing Down Lemma. We will finish with some applications to the structure of families of sets. In particular the \(\Delta\)system Lemma for finite sets. Lecture Notes: before the lecture; during the lecture. Homework Sheet #9. PDF Group Interaction Sheet #8. PDF 
Tenth Lecture: 15 November 2021 
Lecturer: K. P. Hart. Tenth Lecture. Three combinatorial theorems:
Homework Sheet #10. (PDF) Group Interaction Sheet #9. (PDF) Understanding & Explanation Sheet #5. PDF; the same, but with a solution. 
Eleventh Lecture: 22 November 2021 
Lecturer: K. P. Hart. Eleventh Lecture. We start our treatment of forcing. This time we deal with models of Set Theory, or better: structures for the language of Set Theory. The key concepts are absoluteness of notions (so that we do not have to worry when computing simple things like ordered pairs, unions, etc.), and reflection. This enables us to formulate a strategy for proving that a statement like the Continuum Hypothesis is not provable: given finitely many axioms build a countable structure that satisfies those axioms but not the Continuum Hypothesis. This lecture establishes how to do the first half. Lecture Notes: before the lecture; during the lecture. Homework Sheet #11. (PDF) Group Interaction Sheet #10. (PDF) 
Twelfth Lecture: 29 November 2021 
Lecturer: K. P. Hart. Twelfth Lecture. Forcing II: how to extend a model to get a model of the negation of \(\mathsf{CH}\). How to extend a countable transitive set that satisfies (a large portion of) \(\mathsf{ZFC}\) to a similar set that also satisfies \(\neg\mathsf{CH}\). Finite approximations to an injection of \(\omega_2\) into \(\mathcal{P}(\omega)\). An application of the \(\Delta\)system Lemma to show that \(\omega_1\) and \(\omega_2\) are preserved. Lecture Notes: before the lecture; during the lecture. Homework Sheet #12. (PDF) Group Interaction Sheet #11. (PDF) Understanding & Explanation Sheet #6. PDF; the same, but with a solution. 
Thirteenth Lecture : 6 December 2021 
Lecturer: K. P. Hart. Thirteenth Lecture. Today we shall see how \(M[G]\) is constructed from \(M\) and \(G\), and how statements about \(M[G]\) can be translated to statements that `the people in \(M\)' can understand and prove. Generic extensions via arbitrary partial orders: \(M\)generic filters, names, evaluations of names. The definability of forcing and the Truth Lemma. The proper definition of the relation that was used last week. Lecture Notes: before the lecture; during the lecture. Homework Sheet #13. (PDF) Group Interaction Sheet #12. (PDF) 
Lecture Fourteen: 13 December 2021 
Lecturer: K. P. Hart. Fourteenth Lecture. We summarize what we did thus far and show that \(M[G]\) satisfies \(\mathsf{ZFC}\). We finish our discussion of the model of \(\neg\mathsf{CH}\) and calculate the values of \(2^{\aleph_0}\) and \(2^{\aleph_1}\) there (both are equal to \(\aleph_2\)). We shall see how to increase the value of \(2^{\aleph_1}\) without changing \(2^{\aleph_0}\). Lecture Notes: before the lecture; during the lecture. John A. Osmundsen, 2 key mathematics questions answered after quarter century, New York Times, 14 November 1963. Homework Sheet #14. (PDF) Group Interaction Sheet #13. (PDF) 
Fifteenth Lecture: 20 December 2021 
Lecturer: K. P. Hart. Fifteenth Lecture. About the Power Set Axiom and two examples of forcing and consistency results.
The notes contain two more examples of forcing arguments:
Lecture Notes: before the lecture; during the lecture. 
Set Theory: Exam & Resit 
The written exam for the course Set Theory will take place on Monday 17 January 2022 from 2pm to 5pm. The resit exam will be on Monday 21 February 2021 from 2pm to 5pm. Both the exam and the resit exam will take place as an online exam. It is fully open book. 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 and you will have to confirm that you did not do this in writing at the end of the exam. You download the exam from this website at 2pm on the day of the exam, do your handwritten solutions, working until 5pm, and upload them as a single pdf file via this website until 5:15pm. The first two pages of the exam sheet contain some practical information: please make sure that you read them in advance and that you make sure that you can write the exam in an appropriate location: Instruction pages of the exam. (The emergency phone number will be different for the resit.) EXAM STRUCTURE.
We used a similar exam structure last year and you can find last year's Template Exam here: pdf file. You can find comments on the template exam here: pdf file. The exam will resemble the template in structure and style. In particular, the mandatory Part I questions will be Understanding & Explanation questions similar to the questions from the sheets U1 to U6. Comment on Question I.2. Question I.2 asked you to decide in which order you need to force in order to obtain a particular result. This was discussed in Lecture XIV; an unexpectedly large number of students picked the strategy that does not work. We recommend having a look at the discussion of this phenomenon in Kunen's 1980 book (pp. 216 & 217, paragraph marked in yellow). 
Rudiments: Exam & Resit 
The written exam for the Rudiments will take place on Monday 20 December 2021 from 5pm to 7pm. The resit exam will be on Friday 26 March 2022 from 3pm to 5pm. Both the exam and the resit will be 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. 