M285A: INTERMEDIATE FORCING

LECTURE: M-W-F  2:00 -- 2:50   RH 440R

DISCUSSION: Do the following problems from "Forcing Exercises" in the given order: 1.6, 2.1, 4.1, 4.2, 3.1, 3.2, 3.3, 4.3, 4.4 (The argument in 4.4 is similar to that in 3.3, but is slightly more involved.)

COURSE INFORMATION:  This course builds on the first course on set theory and forcing M28. It assumes basic knowledge of forcing, infinitary combinatorics and cardinal arithmetic, large cardinals and definability theory. Below are summaries of knowledge in large cardinals and definability theory you should review, if you don't know them, when we get to use them. I plan to cover the following topics.
1. Two step iteration.    X
2. Regular embeddings and projections.   X 
3. Solovay model.
4. Vopĕnka algebra.
5. Generic ultrapower, precipitous ideals and the duality theorem.
6. Iterated forcing. Factor lemma.
7. Finite support iteration and Martin's Axiom. (This is actually a repetition, meant as a warm-up.)
8. Easton support iteration. Applications: Failure of SCH, Laver's indestructibility.
9. Full support iteration: Club shooting.
10. Proper forcing.
11. Countable support iteration of proper forcing.
12. The proper forcing axiom.
If time permits, more advanced topics will be discussed.

LITERATURE: There is no single piece of literature that would cover all topics in the way we will discuss them. Here is a list of sources which can be viewed as references.
A. Kunen, K.: Set Theory: An introduction to independence proofs, North-Holland 1983
B. Jech, T.: Set Theory,  Springer 2002
C. Cummings, J.: Iterated forcing and elementary embeddings, Handbook of Set Theory, Springer 2010
D. Foreman, M.D.: Generic elementary embeddings, Handbook of Set Theory, Springer 2010
E. Abraham, U.: Proper forcing, Handbook of Set Theory, Springer 2010

BACKGROUND, HOMEWORKS and EXERCISES:  Here are some notes on these topics, I will be posting more throughout the quarter.
(a) I expect that you work throughout most of the "Forcing Exercises" by the end of the quarter.
(b) If there are suitable homework problems I will be posting them along the way. These will be less demanding than "Forcing exercises", and will be taylored to practice technical skills with the material we will be disccussing. In other words, these will require less creativity, although the technical level may be still nontrivial.
(c) I will be posting some notes on background material when I feel it is important.  I will assume you will go over the material (possibly with help of the  literature) to gain  familiarity and technical skills. 

FORCING EXERCISES:   Sheet 1    Sheet 2     Sheet 3    Sheet 4   
HOMEWORKS:   HW1  HW2  HW3  HW4  HW5  HW6  HW7  HW8  HW9  HW10 
BACKGROUND MATERIALS:     Ordinal Definability    Ultrapowers    Extensions of embeddings   ("Extensions" written by Andres Forero)    

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Last update: 2012-05-01