Math 760 (MWF 2:30 -- 3:20 PM, LeConte 316)
Spring 2006 Theory
Set Theory

Instructor: Prof. Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @

Office hours starting April 24:
Monday, April 24: 9-11, 1-2
Wednesday, April 26: 9-10:30
Thursday, April 27: 9:20-11:20, 12:30-1:30
Friday, April 28: 9:30-11:30, 1-3 [tentative]
or by appointment (or you could try dropping by my office at other times).

Textbook: Introduction to Modern Set Theory , by Judith Roitman. Those who have not purchased this book for themselves are receiving photocopies of the book in installments for a fee to cover photocopying costs. Permission has been granted by the author, who holds the copyright.

Last semester, a course announcement was posted to this website telling a bit about what modern set theory is all about, and including a link to an article on the subject by Prof. Nyikos. Click here for that article, published electronically by Topology Atlas, and here for the course announcement posted last semester, minus the paragraph updated here:

On Wednesdays at 3:30 in LeConte 312, starting on January 18, and ending on April 19, there was a seminar in set theory and logic where we went into greater depth on some of the topics mentioned in the course announcement and many more.

Homework problems are whimsically rated G, PG, and R depending on degree of difficulty. Unless otherwise indicated, all problems are rated G and all G-rated problems are worth 5 points, while PG-rated problems are worth 10 points and R-rated problems are worth 20 points. R-rated problems are like extra credit: you can make an A in the course without doing any of them but they can help pull your grade up.

First homework assignment, due Wednesday, January 18:

1. Let # be a preorder on a set X. Define x~y iff x#y and y#x both hold. Show that ~ is an equivalence relation. [Note: I used different symbols, which I don't know how to put into html.]

2. Do Problem 1 on p. 20.

3. Do Problem 2 on p. 20, also answering these questions: What if we put N^+ for the second N? What if we put the real line in place of the first N? Justify your answers.

4. Do Problem 5 on p. 20.

5. Do Problem 6 on p. 20.

6. Do Problem 7 on p. 20.

Second homework assignment, due Monday, January 23:

7. If f and g are functions from omega to itself, define f <* or _ g to mean that f(x)> g(x) for only finitely many x. This is a preorder on the set   X   of all functions from omega to omega. With ~ as in Problem 1, let [f] denote the equivalence class of   f   wrt ~. Show that the following is a well-defined partial order on the set of equivalence classes: [f] < [g] if f <* or _ g but   g is not <* or _ f.

8. [PG] With   f   and   g   as above, define   f<#g   ("f is sharply less than g") to mean   g   is eventually above   f. That is, f(n) < g(n) for all but finitely many n. Show that if   F   is a countable set of functions from omega to omega, then there is a function   g   such that f <# g for all   f in F. [Note that this shows the set of all functions from omega to omega is uncountable.]

9. Show that the definition of order-isomorphism given in the textbook coincides with the definition given in class.

10. [PG] Do 16 on page 21 and show there is an order-isomorphism between the two sets.

11. Do 17 and 18 on page 21.

Third homework assignment, due Monday, January 30:

12. Show that L({0,1}) has an infinite well-ordered subset.

13. [PG] Determine whether L({0,1}) has an infinite dense subset.

14. Do 21 on page 21.

15. Do 1(a) and 1(b) on page 45.

16. Do 2(a)(a) on page 45.

Fourth Homework assignment, due Monday, February 6:

17. [PG] Show that any model of the pairing axiom with more than one element is infinite. From this and the assumption that   x   is not an element of   x   for all sets   x, deduce that every (nonempty) standard model of the pairing axiom is infinite.

18. Prove Theorem 3(f) on page 28.

19. Do 11 on page 46.

20. [PG] Show that if A_n \subset^* w ( w = omega) and A_n is a subset of A_{n+1} for all   n   in w , then there exists A \subset^* w such that A_n \subset^* A for all   n   in w . [Recall that B \subset^* C means B \ C is finite and C \ B is infinite. ]

Fifth Homework assignment, due Monday, February 13:

21. Do 17 on page 46.

22. Do 5 (b) on page 45.

23. [PG] Let U be a proper ultrafilter on w. Given   f, g: w -> w define f ~= g$ to mean that {n in w : f(n) = g(n)} is in U and f ~< g to mean that {n in w : f(n) < or = g(n)} is in U. Show that   ~<   is a preorder relation and   ~=   is an equivalence relation, and that if U is free (``non-principal'') there are uncountably many equivalence classes. [Hint: look at Problems 1, 7, and 8.]

24. [R] Show that if U is a proper non-principal ultrafilter on   R   there is a countable subcollection {U_n : n \in w } of U whose intersection is empty.

Sixth Homework assignment, due Wednesday, February 22 (postponed from Monday, February 20):

25. Show that 7 = {0, 1, 2, 3, 4, 5, 6} is closed under union in both senses of the expression. [Hints: (a) This is a finite set, so to do it in the first sense, it is enough to do it for 2-element subsets. (b) Look carefully at Exercise 15 a.]

26.[PG] With V_n and X defined as in Example 1 on the February 10 handout, show by induction that V_n is a subset of V_{n+1} for all n.

27. Do exercise 7, page 45.

28. [PG] Do exercise 21, page 47.

Seventh Homework assignment, due Monday, February 27:

29. Do Exercise 1abc on page 64.

30. Do Exercise 2a on page 64, displaying UX and UUX. [Here U stands for the union symbol.]

31. Do Exercise 8 on page 65. Up to 5 extra credit points will be given for a coding and display that make it easy to tell any two subsets of V_4 (= elements of V_5) apart.

32. [R] Prove the Main Theorem. You may use either the formulations of psi and phi given here or the ones in the notes of February 15.
If you find this too difficult, you can get 16 points partial credit by doing it for the formulas for Example 37 and 38, and 8 points partial credit by doing it for these two examples up through n = 3.