## Math 768N (MWF 12:20 -- 1:10 PM, LeConte 316) Models of Set Theory

A description of this course posted during Spring semester is here: Models of Set Theory

Homework due Wednesday, August 30:

1. Arrange all the ordinals in exercises 13 and 15 on page 97 in nondecreasing order, stating which ordinals are in fact equal to each other.
This is a 10-point problem and you get four chances at it, your score for unfinished work diminishing by 1/4th each time.

Homework due Wednesday, September 6:

2. Show (b) and (c) in Theorem 16, page 81. [You might find it easier to do in reverse order, and even to use (c) in the proof of (b).]
This is a 20-point problem, and you get a second chance at it without the point values for unfinished work going down; after that, your score for unfinished work diminishes by 1/4th each time.

Homework due Monday, September 11:

3. [10 points] Do number 29 (c) through (h) on pp. 98-99.
After the 11th, your score for unfinished work diminishes by 1/4th each week.

Homework due Monday, September 18:

4. Let k and l be cardinals, with l infinite and k < l . Show that l^k is the number of k-element subsets of l .
This is a 10-point problem, and you get a second chance at it without the point values for unfinished work going down; after that, your score for unfinished work diminishes by 1/4th each time.

Homework due Monday, September 25:

5. Do number 28 on page 98.
This is a 10-point problem, and you get a second chance at it without the point values for unfinished work going down; after that, your score for unfinished work diminishes by 1/4th each time.

Homework due Monday, October 2:

6. [16 points] Do number 9 on page 143.
After the 2nd, your score for unfinished work diminishes by 1/4th each week.

Homework due Monday, October 9:

7. [12 points] Give a detailed proof that       (2^l)^+   -->   ((2^l)^+   ,   l^+)^2       for all infinite   l.
[This is Proposition 6 on the September 29 handout.]
After the 9th, your score for unfinished work diminishes by 1/4th each week, except where I ask you to provide more details.

Homework due Monday, October 16:

8. [10 points] Do number 8 on page 143, justifying your answers.
After the 16th, your score for unfinished work diminishes by 1/4th each week.

Homework due Monday, October 23:

9. [10 points] Using Axioms 1 through 6 in the Baumgartner notes, work the first exercise in the middle of page 40 in Baumgartner's article.]

After the 23rd, your score for unfinished work diminishes by 1/4th each week.

Homework due Monday, October 30:

10. Let $P_\k$ be the Cohen poset described in the notes for October 13. Show that if $\k$ is infinite then forcing with $P_\k$ gives new imaginary'' subsets of $\omega$.

This is a 10-point problem, and you get a second chance at it without the point values for unfinished work going down; after November 8, your score for unfinished work diminishes by 1/4th each time.
Hint. Actually, $\k$ does not need to be infinite, only nonempty! [But since I said "subsets," not "subset," it's simplest to put $\k$ > or = 2.] For each "real" set A, find a dense set D_A of the Cohen poset that shows that the set "dot A_0" of the October 13 notes is distinct from A itself, then get another D(A) that shows "dot A_1" is also distinct from A.

Homework due Monday, November 6:

11. [12 points] Let kappa be an uncountable cardinal. Show that every collection F of finite sets of cardinality kappa contains a Delta-system of cardinality kappa iff kappa is regular.

After the 6th, your score for unfinished work diminishes by 1/4th each week.

Homework due Monday, November 13:

12. [10 points] Do exercise 4 on page 142 in Roitman's text.

After the 13th, your score for unfinished work diminishes by 1/4th each week.

Homework 13, originally due Monday, November 20, has been canceled

Homework due Monday, November 27:

14. [10 points] Show that every MAD family on omega is uncountable.

After the 27th, your score for unfinished work diminishes according to the following schedule: to Dec. 4, 3/4 credit; to Dec. 11, 1/2 credit; to Dec. 15, 1/4 credit

Homework due Monday, December 4:

15. [10 points] Do exercise 5, page 109 in Roitman's text.

After the 4th, your score for unfinished work diminishes according to the following schedule: to Dec. 8, 3/4 credit; to Dec. 13, 1/2 credit; to Dec. 16, 1/4 credit. \bigskip