Schedule

Lecture Videos

First lecture: 13 September 2021

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.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Second lecture: 20 September 2021

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: Dedekind-infinity, 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.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Understanding & Explanation Sheet #1. pdf file.
Here is a comment on how to approach the the Understanding tasks including an example of a good solution for U1: pdf file.

Group Interaction Sheet #1. pdf file.

Third lecture: 27 September 2021

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. Non-canonicity: 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.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Group Interaction Sheet #2. pdf file.

Fourth lecture: 4 October 2021

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 ZF₀. 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.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Understanding & Explanation Sheet #2. pdf file.

Group Interaction Sheet #3. pdf file.

Fifth lecture: 11 October 2021

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 X². Hartogs-Alephs. 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 epsilon-induction. Transitive closures. Foundation implies epsilon-induction. 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.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Group Interaction Sheet #4. pdf file.

Sixth lecture: 18 October 2021

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 Cantor-Schröder-Bernstein 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 Banach-Tarski paradox. Regularity and singularity. All successor cardinals are regular. Lecture Notes.

Homework Sheet #6. pdf file.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Understanding & Explanation Sheet #3. pdf file.

Group Interaction Sheet #5. pdf file.

Seventh lecture: 25 October 2021

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.
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. 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
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

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
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

Group Interaction Sheet #8. PDF

Tenth Lecture: 15 November 2021

Lecturer: K. P. Hart.

Tenth Lecture. Three combinatorial theorems:

• the Free Set Lemma of Hajnal, with emphasis on the difference between the regular and singular cases.
• Ramsey's theorem, the infinite version, and some limiting counterexamples
• The Erdős-Rado theorem

Homework Sheet #10. (PDF)
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

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)
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

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)
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

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)
Student solutions. The solutions were selected from the students' submissions on the grounds that they were good; they are posted with the permission of the authors. Note that they are student solutions, and so they may include real world phenomena like small (spelling) mistakes. Of course, there are many ways to Rome, and if a solution differs from the one given, it may still be a good solution. In particular, the selected solutions are often on the longer, more explanatory side, because this makes for better learning material. pdf file.

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)
Quick solutions. (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.

• Consistency of "every almost disjoint family of uncountable subsets of \(\omega_1\) has cardinality at most \(\aleph_2\) plus \(2^{\aleph_0}>\aleph_2\)" (a nice application of the Erdős-Rado theorem).
• Tennenbaum's construction of a Souslin tree using finite approximations.

The notes contain two more examples of forcing arguments:

• Consistency, due to Kunen, of "the \(\sigma\)-algebra generated by the rectangles in \(\omega_2\times\omega_2\) is not equal to the whole power set (it does not contain the set \(\{\langle\alpha,\beta\rangle:\alpha < \beta\}\))".
• If \(S\) is a stationary subset of \(\omega_1\) then there is a generic extension in which \(\omega_1\) is preserved and \(S\) contains a closed and unbounded subset
(Devlin)

Lecture Notes: before the lecture; during the lecture.

Set Theory: Exam & Re-sit

The written exam for the course Set Theory will take place on Monday 17 January 2022 from 2pm to 5pm. The re-sit exam will be on Monday 21 February 2021 from 2pm to 5pm.

Both the exam and the re-sit 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 hand-written 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 re-sit.)

EXAM STRUCTURE.

1.  The exam will have two parts, a mandatory Part I and an optional Part II. Part I will have two questions and Part II will have three questions.
2. In order to receive a passing exam score, you must attempt both questions in Part I. Each of them is worth 3½ points for a total of 7 points. A satisfactory answer to both questions in Part I will be sufficient to get a passing score in the exam.
3. In addition, you may choose as many of the three questions in Part II, each of which is worth 1 point. The total exam has 10 points.

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 & Re-sit

The written exam for the Rudiments will take place on Monday 20 December 2021 from 5pm to 7pm. The re-sit exam will be on Friday 26 March 2022 from 3pm to 5pm. Both the exam and the re-sit 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.