Student seminar Constructible Sets (WISM551)

Teacher, Time and Venue, Participants

Teacher is Jaap van Oosten. He can be found at room 5.07, tel. 3305. Email: j.vanoosten AT uu.nl

Participants: Anton Golov, Mark Kamsma, Bart Keller, Mireia Martinez i Sellares, Olle Torstensson, Tristan van der Vlugt.
Listeners: Sven Bosman, Tom de Jong, Jetze Zoethout.

The meetings are on Wednesdays 13:15--15:00, in room 610 (except for March 21), HFG. First meeting: Week 8 (Wednesday February 21, 2018).

Requirements, Learning Goals and Grading

Every student presents material, in a blackboard talk. It is permitted to distribute handouts to the audience. Students work in pairs; each session there will be two presentations of 45 minutes (but in each presentation, leave 5 minutes for discussion).

Additionally, every pair of students formulates a homework exercise, which the other participants do, and hand in to the speaker-pair a week later. The speaker-pair then grades this work and hands everything (including a model solution) to the teacher. The teacher, after examination, hands the grades to the participants. Make also a grading scheme: if an exercise consists of more than one part, tell the students what each part is worth.

In the course of the seminar, every student participates in three presentations in pairs.

Attendance is compulsory.

Learning goals are:
1. Student is able to rework a given text into a coherent and understandable presentation.
2. Student has good understanding of the mathematics in the field of the seminar.
3. Student can formulate relevant and challenging exercises.
4. Student develops collaboration and communication skills.
5. Student learns to give constructive feedback to his fellows.

Your final grade is composed of your grade for the presentation (40%, of which 20% for understanding the mathematics and 20% for communicating it), the formulation and grading of the homework exercise (10%) and your solutions to the other speakers' exercises (50%).

Your grade for the presentation will be based on the following aspects:

Subject Matter of the Seminar

The model of "Constructible Sets" was developed by Gödel in the 1930s; it showed that both the Axiom of Choice and the Generalized Continuum Hypothesis can be safely added to Zermelo-Fraenkel Set Theory ZF. The technique is reminiscent of that of the Incompleteness Theorems.

The "Universe of Constructible Sets" (generally denoted L) has many applications and is a classical piece of Set Theory.

Reading Material

The book Constructibility by Keith J. Devlin, Springer 1984.
Note: there are some issues with the Devlin book: for a mathoverflow discussion, see here. For a paper by Mathias, correcting some flaws, see here.

Schedule

Week 8
February 21: Mireia and Tristan: Chapter I, sections 1--5 (quickly), 6,7. Handout and homework. Model Solution.
Week 9
February 28: Olle and Bart: Chapter I, section 8. Homework. Model Solution.
Week 10
March 7: Anton and Mark: Chapter I, section 9. Handout and homework. Model Solution.
Week 11
March 14: Mireia: Chapter I, section 10. Homework. Model Solution.
Week 12
March 21: Note: we meet in Duistermaat, 5th floor of HFG. Tristan and Olle: Chapter II, sections 1,2. Handout and homework. Model Solution.
Week 13
March 28: Bart and Mark: Chapter II, sections 3,4,5. Handout and homework. Model Solution.
Week 14
April 4: Anton and Mireia: Chapter III, section 1. Handout and homework. Model Solution.
Week 15
April 11: Tristan and Bart: Chapter III, section 2. Homework. Model Solution.
Week 16
April 18: Olle and Mark: the paper "Trees" by Jech (Journal of Symbolic Logic 36 (1971), 1, pp.1--14). If ZFC + "There is no Kurepa tree" is consistent, then so is ZFC + "there is an inaccessible cardinal". Handout and homework. Model Solution.
Week 23
April 25: Anton: Chapter IV, section 1 and part of section 2. Handout and homework. Model Solution.
Week 26
May 16: Grading and evaluation of the seminar.

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