A taste of Set Theory (WIS, 2009)

Course's homepage

Sundays 14:00-16:00, Ziskind 1.

Important notices:

  1. In Qn.5 of the last Ex., you do not need to write explicitly what is the formulat pi(x,alpha) telling that rank(x)=alpha.
  2. Due date of the last two exercises (11 and 12) is: Last Friday of the semester break, 6 March 09.
  3. Please write for yourself the email you obtain from yedi777dya@macs.biu.ac.il by removing the string 777. At any time this webpage does not open, please send an urgent message to this email, telling which webpage does not work and asking whether it could be fixed. This is the email of our system Guru. Often he can fix things in minutes.

General instructions

Normally, exercises are posted here each week on Tuesday the latest, and the solutions should be handed on the next Tuesday the latest, in my mailbox.

Solutions can be written in Hebrew or in English, by hand or by computer. In the former case, use a clear handwriting.

To get the maximal grade, all exercises should be handed, and obtain the maximal grade. Any partial handing or no handing will reduce the final grade. Experience shows that all students who did not retire the course obtained a final grade at least 80, but we cannot commit on the future.

Recommended reading

Click here for the Hebrew booklet Cantor's Dream, as well as the details of the book by Kunen (used for the first 1/3rd of the course) and by Jech (whose first part contains the material for most of the course). Both books are available in the library.

I will post scanned parts of these books here, for personal use, in the future. (If I remember to do so.)

Exercises

Frequency. Exercises are given each week. I will try to make the exercises availabe on this webpage each Tuesday, except for today...

The first part of the course is according to the first part of Kunen's book, available in the library.

  1. Exercise 0: Read from the start, until page 13 (not including Cartesian products etc. on the last page) in the book. (No written exercises this time.)

  2. Exercise 1. Click the link on my left. See also more pages from the book (read there until page 15, inclusive).

  3. Exercise 2. Click the link on my left. The required material is covered in the book chapter posted in Exercise 1.
    Good news for those who prefer another style of exposition: Try this.

  4. Exercise 3. Click the link on my left.
    (Error in question about division corrected. Probably you guessed it right anyway.)

  5. Exercise 4. Click the link on my left.

  6. Exercise 5. Click the link on my left.
    Read the full proof that kxk=k.

  7. Exercise 6. Click the link on my left.
    Some more material.

  8. Exercise 7. Read the link on my left, at least pages 51-55, and page 57.
    Solve questions 5.12,5.14,5.18,5.21,5.23 (on pages 60-61), using whatever you need from what you read.

  9. Exercise 8. Click the link on my left.

  10. Exercise 9:
    1. Read pages 73-75 in the following chapter.
    2. Solve Exercise 9.

  11. Exercise 10. Click the link on my left. (Please do not use questions you didn't solve from Exercise 9, in your solution of Exercise 10.)

  12. Exercise 11. Click the link on my left. (Due date: Friday, 6 March 09.)

  13. Exercise 12 (final exercise). Click the link on my left.
    In Qn.5, you do not need to write explicitly what is the formulat pi(x,alpha) telling that rank(x)=alpha. (Due date: Friday, 6 March 09 - same as Ex. 11)

  14. Not Homework: Have a look at the riddle about hats, the infinite case. Prove that the set of representatives chosen there is not Lebesgue measurable.
    Solovay proved that it is consistent with ZF that all sets of reals are Lebesgue measurable. Deduce from this that the solution presented in Wikipedia cannot be obtained without the Axiom of Choice (i.e. in ZF only). [Comment: I just told this fact to a Wikipedia editor. By the time you read this, the answer may already be included there.]

I enjoyed teaching you. Hope to see some of you again in other courses.

Boaz