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# +Axiomatic set theory, spring 2011

**Lecturer**

**Projects: These continue June 17, 12-14, room B322. Nikolay continues his presentation on Boolean-valued models and Hannu discusses diamond. **

**The abstract of paper on the definability of the ground model in the forcing extension is found** **here**** **

**Questions for the FINAL EXAM might be: Prove replacement and power set axioms hold in L. Define absoluteness. Why is "y=transitive closure of x" absolute for transitive M? What can you say about the cardinalities of the V_alpha's, for alpha finite? For alpha infinite? What about the same question for the L_alpha's? Problem 2, a-f, on page 146. Prove theorem 3.5, p. 171, about the minimality of L. Prove theorem 7.5, p. 137. Why is the axiom of choice true in L (sketch the proof in words.) How can countable models of ZFC contain "the reals"? Aren't the reals uncountable?** PLUS: Prove theorem 5.6, 7.5 in chapter 4. Sketch the proof that CH holds in L. Prove that M is a subset of M[G], and that the power set and replacement axioms hold in M[G]. Prove that it is consistent with ZFC that the continuum is exactly = aleph_2. Prove that if the partial order has the ccc-property, then it "preserves cardinals." Plus: one or 2 surprise questions.

Solutions to problem 2, a-f Here

### Scope

10 cu.

### Type

Advanced studies

### Prerequisites: Basic set theory. Cardinal and ordinal arithmetic is needed, but we can review some of this in class.

### Lectures

Weeks 3-9 and 11-18 Mon 12-14, Wed 10-12 in room B322

Easter holiday 21.-27.4.

**We will cover the basic theory of cardinal and ordinal numbers if there is the need to do so. We then go on to constructibility and the consistency of the continuum hypothesis and the axiom of choice. We end with forcing and the consistency of the negation of the continuum hypothesis, the independence of the Suslin hypothesis, and some basics of large cardinals, time permitting.**

### Exams

### Bibliography: Kunen, "Set Theory: An Introduction to Independence Proofs"

### Registration

Did you forget to register? What to do.

### Exercise groups

Group |
Day |
Time |
Place |
Instructor |
---|---|---|---|---|

1. |
Mon |
14-16 |
B321 |
Juliette Kennedy |

**Homework due 24.1**: Read chapter 4. Enderton, page 139: prove theorem 6H; prove theorem 6M, remaining cases. Prove corollary 6P.

**Homework due 31.1:** Prove theorems 7D, 7M, 7N, Hartog's Theorem (p. 195), Well-Ordering Theorem (p.196), Numeration Theorem (p. 197); p. 200, exercises 22, 25. **Homework due 7.2:** Construct the V_alpha's by transfinite recursion on the class ORD. Do the same with the Aleph_alpha's. Prove theorem 7R, 7U, 7W, Enderton, p. 202-205. Prove theorem 8A (Enderton, p.213). (NOTE: On Monday February 7th you will work through these problems together.) **Homework due 10.2:** (This is for the homework or "example class" that meets in an extra session on this day. If this is too short notice for these questions, try to complete as many as you can and we will go over them in class.) Kunen, p. 146: 1,2,3,17. **Note 2.8: We will save problem 17 for next Monday! Homework 14.2:** p. 146, 16, 17. GIve the complete proof of theorem 5.6 on page 129, and 5.4 on page 125. **Homework 21.2**: p. 147: 11,12. **Homework 28.2: For everyone: *p. 146, 2c,2e,2f. (Remember the methods we used in class last Wednesday.) *For those who were assigned these particular theorems**: Prove: p. 166, lemma 1.3; lemma 1.7 on page 156; and the third lemma. **Homework 14.3:** Students continue to present their work. **Homework 21.3:** Exam. **Homework 28.3:** p. 180: 2,3,4,5. **Homework 4.4:** p.180, 4. Prove that the **union,** comprehension, powerset and replacement axioms hold in M[G], where M is a transitive model of ZFC, G is P-generic over M, for some partial order P. Did you need the genericity of the filter? **Homework 4.11:** Prove corollary 3.7 on page 201. Prove König's Theorem. Are there cardinalities, that the cardinality of the reals cannot be? (Hint: Remember König's theorem.) Prove corollary 5.15 on page 209.