## Elements of Set Theory, Spring 2017

**Levy reflection today June 26 at 12:30, 3rd floor blackboards**

**Teacher:** Juliette Kennedy

**Scope:** 10 cr

**Type:** Intermediate

**Teaching:**

**Topics: We will cover ****Dedekind's construction of the reals, moving on from there to ****the basic theory of ordinal and cardinal numbers, equivalents of the Axiom of Choice, and the study of other ZFC axioms. **

**Prerequisites: Some "mathematical maturity" is helpful. Basic logic is helpful too but not required.**

Teaching schedule

Tuesdays and Thursdays 12-14, room C123.

Example class Wednesday 10-12, room DK117.

## Exams

**Your grade for the course is your grade on the final exam.**

## Course material

## We will go through the classic text of H. Enderton called "Elements of Set Theory"

## Registration

## Did you forget to register? What to do?

## Exercises

### Assignment numbers all refer to the textbook.

- Set 1: p. 64: problems 46, 48, 54. p. 70: 1. p. 73: 2,3,4,5,6
- Set 2: p. 78, problem 7. p.83: 13-17.
**IF THERE IS TIME:**p. 88: 18-26 - Set 3: p. 88: 18-26. p. 101: 4, 6,7,8,9
- Set 4: p. 111: 10,11,12,13,14. p. 120: 19,20. Prove: If a continuous function from [0,1] into the reals is negative for some value in [0,1] and positive for some value in [0,1],

then there is a value z in [0,1] such that f(z)=0. HINT: use the least upper bound property. Extra Question: if we define reals as equivalence classes of Cauchy sequences, the how do you prove the least upper bound property (i.e. the completeness of the real line)? Hmmmm....maybe think abut the Axiom of Choice? - Set 5: p. 118, prove theorems 5RH, 5RI, 5RJ. p. 120-121: 16-19. p. 133: 1-5.
- Set 6: Prove the Cantor-Schroder-Bernstein Theorem: If f: A → B is a 1-1 mapping, and g: B→ A is a 1-1 mapping, the the sets A and B are equinumerous. Prove theorem 6I, parts 3,5,6. p. 158: 18-22; 25
- Set 7: p. 178, 4,5,6,7
- Set 8: p. 184, 10,11. p. 187, 12,13,14. Read ahead and do, on p. 194: 16, 18.
**Prove clauses 3 and 4 on p. 181 in the transfinite recursion theorem.** **Set 9: Prove theorem 7I on p. 188. Prove corollary 7N, a,b,c,d. Prove theorem 7M in detail. p. 195, exercises 15,16. p. 199, problem 22.****Set 10: p. 195, problem 18. On page 199, verify the sentence on line 4: "Then D \in C, since D...." p. 200: 23,24,25.****p. 207: 26,27,29, 31,36, 37,38,39**-
**p. 215: 1. p. 219: 3,4,7.****Prove:**If A is a well-ordered proper class, which is "set-like", then A has a least element (with respect to the w.o on A):**Hint:**Consider the formula phi(z) & [forall y ( phi(y)--> W(z,y)], where phi defines the class A and W(x,y) defines the w.o on A. (A w.o. class is "set-like", if seg t is a set for all t in the class.**Corollary:**Transfinite INDUCTION principle (see. p. 174) holds for well-ordered classes. In case the w.o. is not set-like, one can still prove the theorem,but this requires a different proof. See Well-ordered classes.pdf

**LAST HOMEWORK: p. 226: 12-16, 17,18; p. 22,23,24,26**

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## Course feedback

Course feedback can be given at any point during the course. Click here.