Student seminar Models of Intuitionism (WISM549)

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: Menno de Boer, Sven Bosman, Anton Golov, Tom de Jong, Mark Kamsma, Ruben Meuwese, Jan Rooduijn, Bobby Vos, Jetze Zoethout

The meetings are on Thursdays 15:15--17:00, in Duistermaat, HFG. First meeting: Week 7 (Thursday February 16, 2017).

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

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%).

Subject Matter of the Seminar

Intuitionism, a philosophy of mathematics created by the Dutch mathematician L.E.J. Brouwer, rejects the principle of "excluded middle" or "tertium non datur" (if the negation of s statement leads to a contradiction, then that statement must be true) and the Axiom of Choice.

A normal, classical mathematician can only make sense of the intuitionistic philosophy of mathematics via "models" (or interpretations) in the standard mathematical world. Several of these interpretations have been formulated in the course of the 20th century: Kripke and Beth models, complete Heyting algebra-valued models, sheaf and topological models, realizability, interpretations by means of "Kolmogorov problems", interpretations in the "Medvedev lattice", Läuchli semantics, topos semantics.

Actually, interpretations of Intuitionism lead to interesting mathematics.

Reading Material

A.S. Troelstra and D. van Dalen, Constructivism in Mathematics (2 vols), Studies in Logic 121, North-Holland 1988.
J. van Oosten, lecture notes on Category Theory and Topos Theory, revised 2016.
Dana Scott, Topological Interpretation of Intuitionistic Analysis part I (Compositio Math. 20 (1968), pp.194--210), and part II (In: Kino et al (eds), Intuitionism and Proof Theory, North-Holland 1970, pp.235--255).
J. van Oosten, Basic Computability Theory, lecture notes.
S.C. Kleene, On the interpretation of intuitionistic number theory, Journal of Symbolic Logic 10 (1945)No. 4, pp.109--124.
A. Kolmogorov, Zur Deutung der intuitionistischen Logik, Math. Zeitschrift 35 (1932), pp.58--65.
Ju. T. Medvedev, Three papers on finite problems, Soviet Mathematics Doklady 1962--1966.
H. Läuchli, An abstract notion of realizability, in Kino et al (eds), Intuitionism and Proof Theory, North-Holland 1970, pp.227--234.
V.Harnik and M.Makkai, Lambek's Categorical Proof Theory and Läuchli's Abstract Realizability, Journal of Symbolic Logic 57 (1) (1992), pp.200--230.
A.Sorbi and S.Terwijn, Intuitionistic Logic and Muchnik Degrees, Algebra Universalis 67 (2012), 175--188.
V.Lifschitz, CT_0 is stronger than CT_o!, Proceedings of the American Mathematical Society 73 (1) (1979), 101-106.

Schedule

Week 7: Thursday February 16: Ruben and Menno: Kripke semantics. Handout. Homework 1. Model Solution.
Week 8: Thursday February 23 (meeting is in Springer, 7th floor): Bobby and Jan: Beth Models, complete Heyting algebras and $\Omega$-sets. Handout. Homework 2. Model Solution. Grading scheme.
Week 9: Thursday March 2: Mark and Anton: Sheaves and $\Omega$-sets. Handout. Homework 3. Model Solution and grading scheme.
Week 10: Thursday March 9: Sven and Jetze: Scott's topological models. Handout. Homework 4. Model Solution and grading scheme.
Week 11: Thursday March 16: Tom and Ruben: Introduction to computable functions. Handout. Homework 5. Model Solution.
Week 12: Thursday March 23: Menno and Bobby: Kleene realizability. Handout. Homework 6. Model Solution.
Week 13: Thursday March 30: Sven: Kolmogorov and Medvedev Problems. Handout. Homework 7. Model Solution.
Week 14: Thursday April 6: No seminar.
Week 15: Thursday April 13: Mark and Jetze: Läuchli realizability. Handout. Homework 8. Model Solution.
Week 16: Thursday April 20: Tom and Menno: Sorbi and Terwijn. Handout. Homework 9. Model Solution.
Week 17: Thursday April 27: No seminar on account of King's Day.
Week 18: Thursday May 4: Ruben and Bobby: Lifschitz. Handout. Homework 10. Model Solution.
Week 19: Thursday May 11: Seminar starts at 14:15. Tom and Jetze: Makkai and Harnik I. Handout. Homework 11. Model Solution.
At 16:15, professor Martin Hyland (Cambridge) will give a guest lecture in the seminar, on "Models for choice sequences (Fourman, van der Hoeven, Moerdijk) and their relation with Brouwer's original thoughts".
Week 20: Wednesday May 17: Sven and Mark: Makkai and Harnik II. Handout. Homework 12. Model Solution.
Week 23: Thursday June 8: evaluation and grading.

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