Current Office Hours:

Assignment: Read Chapter 1, and work exercises 1-4 on page 6-7, by Friday. (Not to be turned in, at least yet.)

Assignment for Wednesday: Read through page 26.

Exercises: Page 26/1,2,3,4,6,8,9,10
Discuss Wednesday, Jan 21
Underlined problems (only!) due Friday, Jan 23

The following problems should all be review for you (except maybe the last one) so I'm not collecting any:

Page 32/13, 14, 17, 18, 19, 22, 23, 25
Page 34/34, 35 (do this one!)

We'll discuss these Friday, Jan 23

Class: Finish axioms. Algebra of sets (this should be review for you). Start: Relations and Functions (Chapter 3).

Assignment for Friday: Start reading Chapter 3 of Enderton

Assignment for next week: Finish reading Chapter 3 of Enderton

Some homework problems:

p38/1,2, 4, 5a
(Discuss these on Monday, Jan 26)

p65/48, 54, 55, 60
p41/6,7
p42/10
p52-3/11, 13, 15, 18, 21, 23, 28, 30
(Discuss these on Wednesday, Jan 28)

Turn in underlined problems (only!) Friday, Jan 30

Reading: As mentioned above, you should have read Ch. 3 of Enderton, but also you should take a look at the first few pages of Chapter 7 (through the middle of page 172).

Some homework problems:

p. 55/31
(Discuss this on Monday, Feb 2)

p. 61-2/33, 36, 37, 41
p. 64/43, 44, 45
p. 172/2,3 (I know I said in class that we wouldn't do the problems from later chapters until later, but these are easy, and Posets are fresh on our minds, and I wanted to have you turn in at least 3 problems)
(Discuss these on Wednesday, Feb 4)

Turn in underlined problems (only!) Friday, Feb 6

Reading: Read Ch. 4 and 5 of Enderton. I'm not going to say much about "Peano systems"

Homeworks:

p. 70/1
p.73/2-5; turn in 3
p.78/turn in 9-11, think about 12

Discuss Feb 18. Turn in Feb 20

Class: Continue number systems (Ch 4-5).

p.88/20-22, 26.
p.178/5

Don't turn any of these in, but these WILL be covered for the exam.

MIDTERM EXAM: Wednesday, Feb 25

Covers:

• Chapter 1-3;
• Chapter 4 except for the "Arithmetic" section (pp79-82);
• Chapter 7 pp167-178;
• order isomorphism from the lecture notes (it is also in Ch. 7, but that is not readible until we do a few more topics);
• Chapter 9 through p242 (Well founded relations)

Some supplementary/review exercises:

p. 88-89/30,32,34,39 (these should be easy/quick!)

This linked set of extra problems.

Class: Cardinality (Chapter 6)

NOTE: I will be postponing cardinal arithmetic (and the extended discussion of AC) until after we introduce ordinals, becase we need the latter to prove some of the important arithmetic identities.

p.83/14, 17, turn in 13
p.88/23-25
p.88/37, 39, turn in 37a
p. 101/1, 5, 6
p. 111/11, 14
p. 120/19, 21

Even though this is a large set of problems, I want to de-emphasize them, as while the exercises on pp 83-88 are good exercises on induction, none of this material is necessary for understanding any of the stuff we'll be doing moving forward.

p.133/4; turn in 3

These are due on Friday, March 6

Class: Finish countability; start ordinals

Assignment: Review the material from p167-178 (we've already covered this in class!). Pages 180-181 is also material we've covered; I sketched the proof of the Recursion Theorem for in class, the proof in the text of the Transfinite Recursion Theorem follows the same outline but looks much harder because of the additional bookkeeping when working with arbitrary well ordered sets. As we noted in class (and my posted lecture notes), the theorem holds for well founded sets as well. I'm not going to hold you responsible for the proof.

Problems:
p. 138/8,9
p. 161/28, 29; turn in 27 (these are all really good exercises!)
p. 178/turn in 6
p. 182/8, 9

p. 161/28, 29; turn in 27
p. 144/10, 11, 12; turn in 13
p. 150/15, 17
p. 158/20, 21, 22, 24, 25; turn in 21 and 25

SORRY, THIS SECOND GROUP WAS NOT SUPPOSED TO BE ASSIGNED YET!

Discuss March 11; turn in the designated ones March 13

Class: Finish ordinals. Cardinal Arithmetic. Maybe start AC?

We are now 'merging' with the text, and should be through text p.199 by the end of the week, except for the 'equivalents of AC' section. You should read/review this material!

Problems:
p. 144/10, 11, 14; turn in 13
p. 150/15, 17

We can discuss the problems on p144 and p150 this Friday (March 27); the ones to hand in won't be due this week. I am also working on some additional problems.

The text discusses equivalents of AC before proving the absorption laws (p162-165) because they use Zorn's Lemma, but I'm going to prove them by transfinite induction so will put off the AC discussion for a few days.

Problem 13 from p. 144: Turn in April 10.

Class: Absorption Theorem. Start AC.

p. 165/34, 35, 36; turn in 32 and 33 (you might be able to use one to get the other)
p. 158/20, 21, 22, 24, 25; turn in 21 and 25
(Note p158#21 requires AC)

These are due April 10 (together with the p. 144 problem)

Problems from pages 144, 158, 165 are due on Friday

MIDTERM EXAM: Friday, April 17

Covers:

• The "Arithmetic" section of Chapter 4 (pp79-82);
• Chapter 5: you should know the basic definitions of the integers, rationals, and reals, and the exercise from pp 83, 88, 101, and 111 that I assigned the week of March 2. I'm not going to ask you to prove anything about the reals (p. 120 exercises)
• Cardinals and cardinal arithmetic (Ch. 6) , including AC
• Ordinals (Ch 7, Ch 8 through p215)

Approximate breakdown:

15% arithmetic on the natural numbers
15% the integers, rationals, reals
45% cardinals, cardinal arithmetic, AC
25% ordinals

Some supplemental/review exercises:

This linked set of extra/review exercises
p207/35,37,38

Class: Köigs's Lemma, König's Theorem (different König!), Ordinal arithmetic

Lecture notes on Konig's Tree Lemma

Lecture notes on Konig's Cardinal Sum/Product Theorem

Lecture notes on Ordinal Arithmetic

Assignment: We're in Chapter 8, though some of the material (like the Konig Tree Lemma) is not in the text. We've already done through p.215. The fixed-point result (215-219) I'll probably skip. The rest of Chapter 8 (220-240) is ordinal arithmetic, including many proofs I will not be doing in class.

Problems:
p226/14, 15, 19 (this uses 18, which we've discussed in class earlier in the semester)
p250/20, 23, 27

I've decided not to collect these, but at least one will be on the exam.

May 6 is the last day.

FINAL EXAM: Monday, May 11, 12:00-1:59, Keller 301

• The 'usual conditions' apply (open book, open printed lecture notes from class, you'll be allowed to redo a few problems for Wednesday).
• At least 50% of the probems will be drawn directly from the homework.
• You can expect:
  • some problems testing your understanding of the definitions.
    
  • a transfinite induction and/or an induction on a well-ordered or well-founded set.
    
  • some cardinality calculations
• I will ask you to prove one or two 'major' theorems from the course. These include: Cantor's Theorem, The Cantor-Schroeder-Bernstein Theorem, some equivalent of the Axiom of Choice, the absorption theorem for cardinal arithmetic, Konig's Tree Lemma.
• I will ask you a question asking you to synthesize some of your knowledge from the course. For example: How did we use sets to provide a foundation for concrete mathematics? What is the difference between a class and a set, and why do we need classes? How much infinitary cardinality could we still do if we didn't assume AC? (I won't want an essay on this, maybe a 3-4 sentence paragraph.)