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".

Anmäl dig
14.8.2017 kl. 09:00 - 14.12.2017 kl. 23:59


Bild för Juliette Kennedy

Juliette Kennedy

Publicerad, 25.10.2017 kl. 11:19

Dear Students,

Large Cardinals 2 starts next week. The course description is as follows:

"Boban continues the 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. Ehrenfeucht_Fraisse games and the axiom of determinacy are introduced, specifically how large cardinals (such as measurables) yield determinacy. The course is passed by taking an exam, or with a project presented in class."

Remember that the large cardinals course is split into two 5 credit courses, which means you would have to register for the continuation course, Large Cardinals 2, in order to get credit for the full course.

Bild för Juliette Kennedy

Juliette Kennedy

Publicerad, 13.10.2017 kl. 10:33

There will be a class Monday October 16th, from 16-18, room C123.

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Juliette Kennedy

Publicerad, 6.10.2017 kl. 13:06

The pop quiz exam is scheduled for Tuesday 24.10, from 12-15. The info about the room will published on the door of A111 on the day of the exam.

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Juliette Kennedy

Publicerad, 5.10.2017 kl. 18:35

The "pop quiz" exam will take place during exam week (i.e. the week of October 23rd). Place, date and time TBA.

Bild för Juliette Kennedy

Juliette Kennedy

Publicerad, 5.10.2017 kl. 18:33

Revision to schedule the week of October 16: Tuesday will be an ordinary lecture and/or example class. There will also be a class Monday October 16th, from 16-18, room TBA.

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Juliette Kennedy

Publicerad, 2.10.2017 kl. 15:30

The following is our schedule in October:
Week of October 2nd, as usual.
Week of October 9th, Tuesday's lecture class is an example class and or lecture by Hazel. Thursday and Friday lectures by Kennedy.
Week of October 16th: Tuesday will be test day---as discussed in class.

Bild för Juliette Kennedy

Juliette Kennedy

Publicerad, 25.9.2017 kl. 17:16

Hazel Brickhurst will teach tomorrow's lecture (Sept. 26) as an example class. Then we have regular lectures Thursday AND Friday.

We are going to introduce so-called club and stationary sets, moving on to indescribable and other cardinals.

Coincidentally, Hazel is speaking about this in this weeks' Logic Seminar on Wednesday (Sept. 27th), in C124 from 12-14. Hope to see you there!


For bureaucratic reasons, this course is a one quarter course. The next quarter is the continuation, "Large Cardinals 2".

tis 5.9.2017
12:15 - 14:00
tors 7.9.2017
12:15 - 14:00
tis 12.9.2017
12:15 - 14:00
tors 14.9.2017
12:15 - 14:00
tis 19.9.2017
12:15 - 14:00
tors 21.9.2017
12:15 - 14:00
tis 26.9.2017
12:15 - 14:00
tors 28.9.2017
12:15 - 14:00
tis 3.10.2017
12:15 - 14:00
tors 5.10.2017
12:15 - 14:00
tis 10.10.2017
12:15 - 14:00
tors 12.10.2017
12:15 - 14:00
tis 17.10.2017
12:15 - 14:00
tors 19.10.2017
12:15 - 14:00

Övrig undervisning

08.09. - 20.10.2017 fre 12.15-14.00
Juliette Kennedy
Undervisningsspråk: Engelska


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.




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


October 6 homework.

Delta lemma; homework set 4

Homework set 5 DUE TUESDAY OCTOBER 10


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


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


Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later