Office hours and address

Office location: Boyd 603C

Office hours for Fall 2017

  • Mondays: 2:00-3:30pm
  • Tuesdays: 9:30-10:30am
  • Wednesdays: 3:30-5:00pm
  • Thursdays: 9:30-10:30am and 3:30-5:00pm
  • Fridays: 3:30-5:00pm

or by appointment. You can stop by if my door is open.

Mailing address

Dr. Michael Klipper
(706) 542-2588
Boyd GSRC 603C
The University of Georgia
Athens, GA 30602
mklipper@uga.edu

Section Info + Basics

Section 41084 (of Math 4900) is cross-listed with Section 42290 (of Math 6900).

  • It meets MWF 10:10-11:00am in Boyd 322.
  • Syllabus

 

You can find your grades and some solution sets posted on eLC, but that site will serve very little purpose otherwise. Please do not download solution sets to your own computer, but you may print your own copy.

 

General study suggestions for test review

 

Paid tutor options (suggested by the Math department)

  • Since this is a special topic course, it may be harder to find a tutor for this course. You may want to specifically look for graduate students or faculty members in these tutor lists, or go to office hours more.

Math 4900 / 6900, Spring 2017

Class Schedule

 

All textbook sections come from the official course textbook, Introduction to Set Theory by Karel Hrbacek and Thomas Jech. Your homework sets also use many problems from that textbook. For more details, consult the syllabus.

 

Expect this calendar to update weekly. At the end of each week, we will usually have an extra handout which can either show you more examples or can go through a detailed example more thoroughly. Make sure to check this calendar at least on Mondays and Fridays.


UNIT 1

Week 1 and first day of class

Day

Book Sections

Topics / Due Dates

Friday Jan 6

1.3 (skim 1.1 and 1.2) Russell's Paradox, axioms of ZFC (part 1)

Monday Jan 9

1.3

Axioms (part 2), Existence of union and intersection

Wednesday Jan 11

1.4

Several operations, proving set identities

Friday Jan 13

2.1, 2.2

HW 1 due

Ordered pairs, def of relation

Extra handout of the week: Practice with the membership relation and Cartesian products

 

HW 1 is due on Friday, January 13.


Week 2

Day

Book Sections

Topics / Due Dates

Monday Jan 16

 

Martin Luther King Jr Day

no class

Wednesday Jan 18

2.2, 2.3

Domain and range, inverse and composite, images and preimages

Friday Jan 20

2.3

Notation for functions and sequences, one-to-one functions

Extra handout of the week: Indexed collections, characteristic functions

 

HW 2 is due on Monday, January 23.


Week 3

Day

Book Sections

Topics / Due Dates

Monday Jan 23

2.5, 3.1

(You may want to skim 2.4.)

HW 2 due

Partial orders, inductive sets

(Skip the parts in 2.5 about isomorphisms for now.)

Wednesday Jan 25

3.2

Proof that < is linear on N, and well-ordering

Friday Jan 27

3.2, 3.3

Strong induction, the Recursion Theorem

Extra handout of the week: Minimal vs. least members, a "partial recursion" theorem that can end early

 

HW 3 is due on Monday, January 30.


Week 4

Day

Book Sections

Topics / Due Dates

Monday Jan 30

3.3, 3.4

HW 3 due

Enumerating subsets of N, recursive definitions of plus and times

Wednesday Feb 1

3.4

Multiplication, Exponentiation, some more recursion practice

Friday Feb 3

4.1

Comparing cardinality |A| = |B| and |A| <= |B|, Cantor-Bernstein Theorem

Extra handout of the week: Strong induction practice, and basic properties of |A| <= |B|

 

HW 4 is due on Monday, February 6.

Test 1 is next Friday, February 10. (It lasts 55 minutes, not 50!)

  • You'll find review material uploaded in next week's schedule. This includes a topic list and a practice set of questions!
    • Solutions to those practice questions will be made available as well.
  • In-class review for the test is Wednesday, February 8.

UNIT 2

Week 5

Day

Book Sections

Topics / Due Dates

Monday Feb 6

5.1

HW 4 due

Basic cardinal arithmetic

Wednesday Feb 8

 

Test 1 Review

Friday Feb 10

 

TEST 1 (55 minutes)

 

Test 1 Review Information:


UNIT 2

Week 6

Day

Book Sections

Topics / Due Dates

Monday Feb 13

4.2, part of 4.6 Finite cardinality, showing |P(S)| = |2^S|

Wednesday Feb 15

4.3

Countable sets

Friday Feb 17

4.4

Isomorphic orderings, and a uniqueness property of Q

Extra handout of the week: The famous diagonalization argument, and |N^N|

 

HW 5 is due on Monday, February 20.


Week 7

Day

Book Sections

Topics / Due Dates

Monday Feb 20

6.1

HW 5 due

Properties of well-orderings, existence of sups (completeness), initial segments W[a]

Wednesday Feb 22

6.2

Definition of transitive set and ordinal, showing alpha in beta iff alpha is a subset of beta

Friday Feb 24

6.2, beginning of 6.4

Main ordinal properties, principles of Transfinite Induction

Extra handout of the week: More isomorphism practice, and exploration of limit ordinals

 

HW 6 is due on Monday, February 27.

  • This was edited on Thursday February 23 to fix #4.

Week 8

Day

Book Sections

Topics / Due Dates

Monday Feb 27

6.4, though briefly look at Theorem 6.3.6

 

HW 6 due

Proper classes and operations, Replacement Scheme, Transfinite Recursion Theorem

Wednesday Mar 1

6.4, 6.3

Versions of recursion, enumerating any well-order, Mostowski collapse

Friday Mar 3

Not really in the book, but related to Sections 6.4 and start of 6.6

More examples, increasing continuous operations (leading up to Sections 6.5 and 6.6)

Extra handout of the week: Well-ordered recursion, and a proof by transfinite induction

 

HW 7 is due on Monday, March 13, after Spring Break.

  • However, it is a shorter assignment, and it's only based on class material up through Wednesday. You could try to finish it by Friday this week!

Week 9

Day

Book Sections

Topics / Due Dates

Monday Mar 13

6.5

HW 7 due

Definition of ordinal sum, isomorphism to W_1 + W_2

Wednesday Mar 15

6.5, 6.6

Ordinal product, simplifying some calculations with ordinals, quotients and remainders

Friday Mar 17

6.5, 6.6

Isomorphism to W_2 x W_1, ordinal exponentiation, omega-normal form

Extra handout of the week: Sups and infs, and designing subsets of Q with certain order types

 

HW 8 is due on Monday, March 20.

 

 

Test 2 is next Friday, March 24. (It lasts 55 minutes, not 50!)

  • You'll find review material uploaded in next week's schedule. This includes a topic list and a practice set of questions!
    • Solutions to those practice questions will be made available as well.
  • In-class review for the test is Wednesday, March 22.

Week 10

Day

Book Sections

Topics / Due Dates

Monday Mar 20

6.6

HW 8 due

Addition and multiplication in base omega form, Goodstein sequences

Wednesday Mar 22

 

Test 2 Review

Friday Mar 24

 

TEST 2 (55 minutes)

 

If you're curious about more information on Goodstein sequences, check out this blog posting and this applet to compute Goodstein sequences!

  • The applet is a little clumsy when I tried it out, so the initial number you try might not work quite like you want, but you should still be able to click "Calculate" repeatedly in that applet and see some cool behavior. You may also have to reload the page to try a new number.

 

Test 2 Review Information:


UNIT 3

Week 11

Day

Book Sections

Topics / Due Dates

Monday Mar 27

7.1

Initial ordinals / cardinals, the Hartog's number and the aleph hierarchy

Wednesday Mar 29

7.1, 7.2

The main theorem about sum and product of two infinite cardinals

Friday Mar 31

7.2, 8.1

Some more cardinal simplification, discussion of well-ordering and choice functions

Extra handout of the week: Well-orderable sets, cardinality operations with sets of real numbers

 

HW 9 is due on Monday, April 3.


Week 12

Day

Book Sections

Topics / Due Dates

Monday Apr 3

8.1

HW 9 due

Axiom of Choice, Zorn's Lemma

Wednesday Apr 5

8.1, 8.2

Equivalent versions of Zorn's Lemma, some misc applications of AC / Zorn (not tested)

Friday Mar 31

9.1, 9.2

Sum and product of a collection of cardinals, introduction to cofinality of limit ordinals

Extra handout of the week: Tukey's Lemma about finite character, and building a non-measurable set of reals

Also, watch this for more information on the Banach-Tarski Paradox. (Thank you to a student who shared this!)

 

HW 10 is due on Monday, April 10.


Week 13

Day

Book Sections

Topics / Due Dates

Monday Apr 10

9.1, 9.2

HW 10 due

Regular and singular cardinals, cofinality results, Konig's Theorem

Wednesday Apr 12

9.2, 9.3, 14.1

A couple results about kappa^lambda and cofinality, well-founded relations

Friday Apr 14

14.1

Well-Founded Induction and Recursion, revisit the Mostowski Collapse

Extra handout of the week: Continuous operations and cofinality, well-founded classes and "set-like" relations

 

HW 11 is due on Monday, April 17.

  • This is the final HW set of the course!
  • On Monday April 17, we have one more class to finish Section 14.2. This is not going to be on Test 3.
  • After Test 3, we'll just have one last class with new material (not testable on the final), and then one day of final exam review!

 

Test 3 is next Friday, April 21. (It lasts 55 minutes, not 50!)

  • You'll find review material uploaded in next week's schedule. This includes a topic list and a practice set of questions!
    • Solutions to those practice questions will be made available as well.
  • In-class review for the test is Wednesday, April 19.

Weeks 14 and 15

Day

Book Sections

Topics / Due Dates

Monday Apr 17

14.2

HW 11 due

The V_{alpha} sets, well-founded sets, Axiom of Foundation

Wednesday Apr 19

 

Test 3 Review

Friday Apr 21

 

TEST 3 (55 minutes)

Monday Apr 24

Parts of 10.1, 4.5

Conclusion: where do numbers come from? Dedekind cuts

Wednesday Apr 26

 

Final Review

 

Test 3 Review Information:


Final exam information

Date and time:

  • Monday 5/1 from 8am to 11am in Boyd 322 (our usual classroom)

 

Office hours for finals:

  • Wednesday April 26: 2:00pm to 4:30pm
  • Thursday April 27: all day (8am to 5pm), but try to email in advance
  • Friday April 28: 3:30pm to 5:30pm
  • Sunday office hours may be possible if several people request them by email.

 

Expectations for questions on the final exam:

  • The final should be about twice the length of an in-class test, so expect about 10-12 questions for 200 points in total.
    • There will be one or two extra-credit challenge questions at the end.
  • One question will cover definitions. There will also be a question asking for short examples with no proofs required!
    • For example, for "a non-linear partial ordering", possible answers include "P(omega) ordered by subset" or "N ordered by divisibility |".
  • Most questions should be similar in difficulty and scope to previous test problems: expect about 2-3 problems per unit of class material.
    • However, some questions may incorporate material from several units.
    • Study your old tests!
    • Any topics that were off-limits from earlier tests are still off-limits on this final exam.


Last updated: 4/25/17