SEPTEMBER 5: We refreshed our knowledge of the basics of the theory of cardinals and ordinals, up through the construction of the V_alpha's. The aim in the near future is to finish Enderton's book, which we will do in the next 3 lectures, approximately.

SEPTEMBER 8: We covered transfinite INDUCTION and transfinite RECURSION theorems in Kunen, i.e. theorems 9.2. and 9.3 on pages 24,25. We also defined ordinal addition by trans. recursion.

September 12th: We proved that every cardinal number is an Aleph_alpha, by transfinite induction. We then identified, for a ZFC axiom phi, for which alpha does V_alpha satisfy phi. All axioms were treated except replacement.

Sept. 14: We showed that V_kappa, for kappa inaccessible, is a model of all the ZFC axioms. Introduced the concept of cofinality.

Sept. 19: More on cofinality! We proved: cof(kappa) is always a regular cardinal; if lambda is a limit ordinal then cof(lambda)=cof(Aleph_lambda); König's theorem.

Sept. 21: We proved König's theorem again, clarifying the "sticky point". (By the way, did we use the axiom of choice in the proof?) We did some cardinal arithmetic, using cofinality. (See homework set 3 for the proof.)

Sept. 28 We defined "club subset of \kappa". We proved that the intersection of 2 club sets is club; we proved that the intersection of fewer than \kappa club sets is club. We did another cardinal exponentiation calculation, under certain cofnality assumptions; we (informally) discussed some definitions of large cardinals Mahlo, weakly inaccessible, measurable. For the material on club and stationary sets, we rely on Jech's book Set Theory (3rd ed.), chapter 8.

Sept. 29th: After review of the basics of clubs, we proved that the diagonal intersection of club sets is a club.

October 3,5: We introduce the concepts of Mahlo and weakly compact cardinals. We proved Fodor's lemma, and then we proved the delta lemma using Fodor's lemma; we showed/will show that weakly compact cardinals are inaccessible and have the tree property; if a cardinal is inaccessible and has the tree property, then it is weakly compact. We proved that \omega has the tree property.

Last lectures of the semester: Existence of an Aronszajn tree on \omega_1; Background on the measure problem;
Definition of two valued measure and ultrafilter, showing they are essentially the same; Definition of measurable cardinal; Proof that a measurable cardinal is regular, inaccessible, weakly compact; Definition and brief discussion of L;
Proof of Mostowski collapse; Definition of ultrapower of V; Los theorem; Proof that if there is measurable cardinal then there is an elementary embedding from V into an inner model with critical point the measurable; Scott's theorem