When scheduling the session during the summer camp we planned to organise it during the long November weekend - this great, but difficult branch of mathematics needs peace, focus and time, so this extra day was meaningful. Because the holiday resort in Szczyrk was already taken, we went to the - already known to most of us - hostel in Ustroń.
Our President arrived with provision a day earlier, so everyone (including guests from all Poland) were welcomed with warm and energizing lunch (baked beans in tomato sauce, so you know what I mean). We began working almost immediately - we began the common room's occupation (the rest of the hostel guests did not like it very much). After a short organisational introduction, a brief historical overview of our topic was given by Szymon and then Michał Stolorz began his talk (we had plenty of Michałs), where he introduced the ZF and AC axioms and the notion of an ordinal number. Roksana Policzew developed the topic, while changing the subject to cardinal numbers. Rafał kept on nitpicking and at the end he showed us a few (heh heh, a few) countable ordinal numbers.
On Friday we spent 5 good hours walking - Trzy Kopce Wi¶lańskie were conquered by us and we have seen the hometown of Adam Małysz. After lunch (traditionally - spaghetti) Agnieszka Klama told us about the regularity axiom. During Monika's talk, who presented how does Zorn's Lemma (so AC, indirectly) implies that every vector space has a basis, a discussion emerged about the essentiality of certain assumptions in the vector space's definition. By the way - it is a long time since we last had a session with an algebraic topic. Michał Seweryn was supposed to talk about the Hahn Banach theorem, but he thought it to be too simple, so he showed us some statements equivalent to AC (and he proved some implications). After dinner Michał was also quite a good psychologist (again - an information for the initiated).
Janek Kontrymowicz began on Saturday: we got to know that CH implies the existence of a set on the plane, whose all vertical cuts are finite and horizontal cofinite. While discussing the indicator function of such a set it was easy to see the essentiality of certain assumptions in the Fubini theorem. During the aftersession discussion we quickly constructed a function whose iterated functions were equal with the function itself being non-measurable, so it was an example proving that the Fubini theorem can not be reversed. Janek Boroński explained us the "plus" in the topic of the session by venturing a little bit in the direction of topology: while the topic was set-theoretical - Independent families of sets - the methods were topological in nature. After him Michał Lewicki (the third Michał and it is not the end!) made an introduction to the Model Theory and prepared the ground for Anna Wojciechowska, who splendidly explained to us the problem of constructibility of sets. We got back to the talks after a lunch break (beetroot soup). Szymon explained what Martin's Axiom is all about and showed us a few of its consequences - among others referencing the last year's session about measure and categories. The session was ended with the talk "Big cardinal numbers and ultrapowers" by dr Michał Machura. We got to know what are measurable numbers, (strongly) inaccessible cardinals and why one should (not) be interested in them. We didn't manage to organise the ping-pong tournament, but we still played. Darts was more popular though. Of course, we had great salads, as usual.
last update: 01.06.2012