Kaisa_2012_3_photo by Veikko Somerpuro
Large Cardinals
Large cardinals, trees, combinatorics, ultrafilters and other assorted topics.
Co-taught during November with Boban Velickovic.
We will finish the last chapters of Enderton's book that we did not get to last semester. We then will study the combinatorics of "small" large cardinals such as the weakly compact ones, their equivalent definitions in terms of trees, filters, ultraproducts, Martin's Axiom, reflection and other assorted topics TBA. We will review the material on Ramsey Axioms started by Boban Velickovic in his tutorial last June. The exercise class will be partly used for a review of the logical basics and will be co-taught with post-doc Hazel Brickhill. Forcing is not required for this course but the course Elements of Set Theory, or equivalent knowledge, is required. The course is passed by taking an exam, or with a project presented in class.
For bureaucratic reasons the course is a one quarter course of 5 credits, with the continuation Large Cardinals 2, of 5 credits, being taught in the second quarter. All lectures on room Exactum, C129.
For homework see the link "tasks" below. To see what we covered in the lectures, see the link "what we covered in the lectures".
Meddelande
Tidsschema
Material
Enderton's text called Elements of Set Theory, plus Kunen's Set Theory: An Introduction to Independence Proofs are the main text of the course. However not all subject matter is covered in these texts.
SPECIALIZED COURSE MATERIAL WILL BE POSTED AS THE COURSE PROGRESSES.
Övriga
Uppgifterna
Example class Sept. 8. 2 questions:
1. Prove that is kappa and lambda are 2 infinite cardinals, kappa x lambda = kappa + lambda = max(kappa, lambda)
2. Construct the V_alpha's "piecewise", by transfinite recursion on a large enough delta. Then contract them in one go, all at once.
TOMORROW SEPT: 8th example class
This will be an ordinary lecture. And a very important one! As tomorrow we start using Kunen's book Axiomatic Set Theory.
SEPTEMBER 15 homework:
1. Prove, in Kunen's theorem 9.3 (transfinite recursion):
A. Uniqueness: if G_1 and G_2 both satisfy: for all alpha, [ G (alpha) = F ( G restricted to alpha ) ] , then for all alpha, G_1(alpha)=G_2(alpha).
B. As in the uniqueness proof, if g is a delta-approximation and g’ is a delta'-approximation, then g and g' agree on the restriction of their domain to delta intersect delta'.
2. Construct ordinal exponentiation by transfinite recursion on Ord. What is omega^(omega ^ omega)?
September 22 homework
Prove theorem 9J on page 254, Enderton. On Enderton page 256, 257, do: 5,6,9,11
September 26 homework
SEPT. 26 HOMEWORK
October 6 homework.
Delta lemma; homework set 4
Homework set 5 DUE TUESDAY OCTOBER 10
WHAT WE COVERED IN THE LECTURES
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.
Beskrivning
Optional course.
Master's Programme in Mathematics and Statistics is responsible for the course.
The course belongs to the Mathematics and Applied mathematics module.
The course is available to students from other degree programmes.
Set Theory
Master studies
Basic knowledge of the theory of large cardinals, their equivalents in terms of tree properties, compactness etc
Recommended time/stage of studies for completion: 1. or 2. year
Term/teaching period when the course will be offered: varying
Large cardinals, trees, compactness properties
Lecture notes
Lectures
Exam and excercises, Course will be graded with grades 1-5
Exam, other methods will be described later