Set theory, reading course

Kurt Gödel Research Center, University of Vienna

Lecturer: Yurii Khomskii
Time & Location: Wednesdays at 15:15, in the KGRC Lecture Hall

Credits: 7.0 ECTS

Course content: This course will continue approximately at the point where the course Axiomatische Mengenlehre 1, taught by Jakob Kellner, left off. We will spend the first 2-3 weeks revising MA and the method of forcing. Then we will continue with the book "Set Theory" by Kenneth Kunen, covering more advanced applications of forcing, particularly (most of) the last chapter of Kunen on iterated forcing and the consistency of MA. After that we will study proper forcing and iterations of proper forcing (following texts other than Kunen), and, depending on the time, other applications of the forcing method.
Literature: The primary study material is the book
Kenneth Kunen Set theory, Studies in Logic (London), 34. College Publications, London, 2011. viii+401 pp. ISBN: 978-1-84890-050-9.

We will certainly follow it in the first part of the course. Later on we might switch to other reading material, which will be announced on this website.
In addition to the obligatory reading material, I will often post supplementary material, dealing e.g. with philosophical or historical issues not covered by the standard material, alternative explanations which might help understanding certain concepts, or some other additions making the course more fun.
Format: This is a reading course, which means that the students are expected to read a certain section of the material every week. The weekly meetings consist of two blocks of 45 min., with a short (~5 min.) break. Generally, I will use the first block to "lecture" and the second to discuss questions from student, do practice exercises etc. In the "lectures", I will present a summary of the material the students had to read, distilling the most important concepts and perhaps presenting things in a slightly different way, though not going into all the details (so just following the lectures without the reading is not sufficient to learn the material; the lectures are only of secondary importance). In the second half, I would encourage students to come up with questions, for example about parts of the material they haven't understood, or we will do practice exercises related to the material.

Weekly material

Meeting	Date & Time	Material to be prepared	Section and Exercises	Supplementary material

1	Wednesday 9 October	Revision of MA Revision of "generic extensions" Metamathematics of forcing	Kunen III.3 until/incl. Lemma III.3.15 and Definition III.3.19 - Lemma III.3.20 Kunen IV.1 until/incl. Lemma IV.2.12 Kunen IV.5 Exercises: IV.2.8	Paul Cohen, The Discovery of Forcing. Very insightful and interesting paper (based on a talk) by Paul Cohen, recounting the history of the forcing method and how he "discovered" it.

2	Wednesday 16 October	Definition of "semantic" forcing relation \|\|- Truth and Definability Lemmas without the proofs M[G] \|= ZFC (assuming the above)	Kunen: Lemma IV.2.12 until/incl. Lemma IV.2.32 Exercises: Prove Lemma IV.2.14 Exercise IV.2.16 Exercise IV.2.28 Exercise IV.2.46

3	Wednesday 23 October	The "syntactic" forcing relation \|\|-* Proofs of Truth and Definability Lemmas Equivalent definitions of "genericity"	Kunen: Rest of Section IV.2 Definition III.3.57 - Theorem III.3.60 (don't worry about linked families) Exercises: Exercise IV.2.47 Exercise (A4), (A5), (A6), (A7), (A8) and (A12) from Kunen's old book.	Akihiro Kanamori, Cohen and Set Theory. Another interesting paper about Cohen and the history of forcing.

4	Wednesday 30 October	Computing cardinal exponentiation Embeddings of posets	Kunen: Section IV.3 (forget Kurepa trees in IV.3.17) NB: you will need III.2.5, III.2.6 and III.3.7. Kunen: Definition III.3.63 - Lemma III.3.67. Kunen: Section IV.4 until/including Lemma IV.3.7. and Definition IV.4.14 - Theorem IV.4.15 Exercises: Challange: try solving Exercise IV.2.47! Exercise IV.3.12 Exercise IV.3.18 Exercise IV.4.16	6-part video recording of a talk by Paul Cohen, right here in Vienna, in 2006. (Put the volume up!) Part 1/6 Part 2/6 Part 3/6 Part 4/6 Part 5/6 Part 6/6

5	Wednesday 6 November	Maximality priniple Forcing with higher cardinalities "Forcing over V"	Kunen: Section IV.7 until/incl. Lemma IV.7.25 Exercises: Exercise IV.7.10 Exercise IV.7.11 Exercise IV.7.19 Exercise IV.7.20 Exercise IV.7.24

6	Wednesday 13 November	Product forcing Application of products	Kunen: Section V.1 and V.2 until/incl. Exercise V.2.16 (you may skip Exercises V.2.9, V.2.10 and V.2.11.) Exercises: Exercise V.2.14 Exercise V.2.15 Exercise V.2.16

7	Wednesday 20 November	General two-step iterations General transfinite iterations	Kunen: Section V.3 Exercises: Exercise V.3.7 Exercise V.3.10 Prove Lemma V.3.12	Timothy Y. Chow, A beginner's guide to forcing. A nice article attempting to present the "intuition" behind forcing, written by a non-logician, primarily for non-logicians, and offering some interesting insights. Section 6 of this paper uses the "Booelan-valued models" approach, which we haven't covered (it is explained on p. 275-278 in Kunen), but you can easily read the rest of the paper.

8	Wednesday 27 November	Consistency of MA An easy application of MA	Kunen: Lemma III.3.51 (Or see Jech on p. 272: Kunen: Section V.3 until/incl. Corollary V.4.7. Kunen: revise definition of "add(null)" from section III.1 Kunen: Lemma III.3.28 (see Definition III.3.16) Exercises: Prove the consistency of κ ≤ add(null) directly, without using MA, for regular κ > ω₁

9	Wednesday 4 December	Repeating consistency of MA Some applications of MA NB: This week, there is very little new material so that we can have more time to study MA and its applications. You can read a slightly different explanation of the proof of MA in Chapter 16 of Jech's book. You can also use this week to revise some of the older material.	Revision of Kunen: Section V.3 until/incl. Corollary V.4.7. Kunen: revise definition of "add(null)", "add(meager) from section III.1, and "dominating" and "bounding" numbers, Definition III.1.11. Kunen: Lemma III.3.28 Chapter 16 of Jech: Theorem 16.23 and Theorem 16.24 (these are analogues of Lemma III.3.28 from Kunen but with "add(null)" replaced by "add(meager)" and "b", respectively). Exercises: Think about proving the consistency of "κ ≤ add(null)", "κ ≤ add(meager)" and "κ ≤ b" directly, without using MA, for regular κ > ω₁.

10	Wednesday 11 December	Revision about club/stationary sets Proper forcing NB: Although Kunen's focus is on PFA, we will not really consider this axiom but rather focus on specific iterations of proper forcing.	Kunen: Definition III.6.22 until/incl. Exercise III.6.27. If you are not familiar with club/stationary sets read the beginning of III.6 until/incl. Lemma III.6.14. Kunen: Section V.7 (introduction), V.7.2. and V.7.4 NB: Ignore references to things we haven't covered (KH, SH, SOCA etc.) Exercises: Exercise V.7.14. Exercise IV.7.59 (in the definition of C', replace "p forces B in C-dot" by "p forces A in C-dot"). Exercise IV.7.61

11	Wednesday 18 December	Finite support iterations and Cohen reals A different presentation of proper forcing Sacks forcing Remarks: "fsi" and "csi" abbreviates "finite support iteration" and "countable support interation", respectively. Theorem 4.3 is a simplified version of one of the directions of Theorem V.7.17 from Kunen, so it might be helpful to understand the proof of V.7.17 better.	We will use these lecture notes by Jörg Brendle. Section 3.6 (p 22) Section 4.1 and 4.2 (pp 23-26) Section 5.1 (pp 30-31)	Wilfred Hodges, An Editor Recalls Some Hopeless Papers. An amusing paper, quite unrelated to anything we are doing in this course, but still fun.

12	Wednesday 8 January	OD and HOD Independence of the Axiom of Choice	Kunen Section II.8 Kunen Section IV.8 Exercises: Exercise II.8.7 Exercise II.8.8 Exercise II.8.9 Exercise II.8.11 Exercise IV.8.8 Exercise IV.8.14.

13	Wednesday 15 January	The Levy Collapse	For the rest of the course, we will use the textbook Akihiro Kanamori, The Higher Infinite. Chapter 3 (Sections 10 - 15). "The Levy Collapse", page 126, until the end of section 9 (page 131), excluding Proposition 10.18 and the paragraph preceding it.

14	Wednesday 21 January	Basic properties of measure and category Random reals Solovay's model	We will also need § 0 of Kanamori § 0, section "Measure and Category" (pp 10-14). Remark: you do not need everything from this section. The only important things are the definitions of "null", "Lebesgue measurable" and Lemma 0.9, and the definition of "meager" and "Baire property". Also, note that while Kanamori presents the results in terms of the Baire space, this is not essential, and you can think of the standard real numbers if you prefer, replacing "basic open sets O(s)" by "intervals with rational endpoints". Chapter 11, p 132: Read the formulation of Theorem 11.1; we are only interested in parts (a) and (b) of the theorem. Skip directly to section "Random Reals", p 136. Continue with section "Proof of Solovay's Theorem", p 139. In the proof of 11.11, skip the part about the perfect set property ("Finally, we verify... This completes the proof of 11.11"). Continue until (not including) Proposition 11.13. Remark: you can replace "L(R)" by "HOD^R".

Final Exam: there will be 20-30 min. individual oral exam. The purpose of the oral exam is to check how much of the material was understood. The focus will be on understanding the concepts and main ideas in the proofs, but not every single detail. For example, if I ask about the proof of the consistency of MA, you should know how iterated forcing is used in the proof, the bookkeeping argument, the use of the finite support etc., but you don't have to be able to reproduce the entire proof during the exam. I will also not ask for things like the technical definition of ||-^*, or some very technical proofs related to products and iterations. Also, certain topics that were not covered particularly well during the course or were a bit too hard, are excluded from the exam. Below is an indication of the material to be prepared.

Material for the exam:

Definition of MA and some easy consequences of MA.

Basics of forcing, forcing relation ||-, Truth and Definability Lemmas etc.

Cohen's basic construction for changing the cardinality of the continuum.

Higher cardinality version of Cohen forcing.

Product forcing and Easton's theorem about changing cardinalities of s^κ for many κ simultaneously.

Iterated forcing, focusing on finite support.

Consistency of MA.

Proper forcing: basic definition (in terms of stationary sets), equivalent definition (in terms of elementary submodels) and proof that proper implies ccc. You don't need to know the proof about the equivalence of the two definitions, or about Sacks forcing.

OD, HOD, HOD^R and the proof of consistency of ZF + non-AC.

Solovay model: the basic properties of the Levy collapse and random reals, and an approximate sketch of the proof, but you don't need to know any details of this proof.