# Axiomatic set theory, fall 2008

### Lecturer

### 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 |