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Axiomatic set theory, fall 2008

Lecturer

PhD Juliette Kennedy

Scope

10 sp.

Type

Advanced studies.

Lectures

Weeks 36-42 and 44-50, Monday 16-18 and Wednesday 16-18 in room B322. For corrections to Lemma 2.8 page 96 see:http://mat-238.math.helsinki.fi/note.pdf

Contents

After covering the basic axioms, we move on to constructibility and the beginnings of forcing. Our text is Kunen's "Axiomatic Set Theory."

Prerequisites

Students should be familiar with the basic theory of cardinal and ordinal numbers, as set out for example in Enderton's "Elements of Set Theory."

Course material

Homework.

Week

Exercises

37

Prove Theorem 7.3, 1-6 in detail. Page 43, problems 4,8,9.

38

Prove Theorem 7.18, 1-5, 7.20, 1-6, 10.6, 10.10 in detail. Page 44, problems 10 a-c,15.

39

p.45, 18. Prove: there is a continuous, strictly increasing cofinal function from cf(alpha) to alpha. (An increasing function f is continuous if f(u) = sup(f(z), z<u) for each limit u.) Prove: f is continuous iff f is continuous in the order topology of ordinals. (I.e. the topology generated by open intervals.) NOTE: STUDENTS DO NOT HAVE TO FINISH THESE BY THE TIME THE EXERCISE CLASS MEETS, BUT THE CLASS WILL BE HELD AND THE EXERCISES WILL BE GONE OVER.

40

This week Lauri will review section 14.3 in preparation for chapter 4.

41

Prove lemma 3.5, page 99. Prove Theorem 3.6, page 100. Prove lemma 5.8, page 104. page 107, exercises 1,3.

43

Page 146, 1,2,3.

44

p.131, prove Lemma 6.4 in detail. p.147, 8, 14, 17.

45

The instructor will discuss HOD and prove p.163, 5.

46

p. 182, 19,20,21.

47

p.180, 1,4,5.

48

p.238,A5-A8. Prove lemmas 2.11-18 in detail, p. 190.

49

Prove theorem 4.2 in detail.

Registration

Final Exam: Monday, December 15, 12-16, B322. NOTE NEW EXAM TIME.

Exercise groups

Group

Day

Time

Place

Instructor

1.

Tue

12 - 14

B312

Lauri Tuomi

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