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Elements of Set Theory, Spring 2017

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.

News  Final Exam is May 16th, 12-16. Room to be announced.

Teaching schedule

Tuesdays and Thursdays 12-14, room C123.

Example class Wednesday 10-12, room DK117. 

Exams 

You can pass this course by taking the final exam at the end of the course. The final exam covers the material of the whole course. The exam lasts 2,5 hours. You can also pass the course with a project.

In class was mentioned a mid-term exam but this will not be given.

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 

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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

 

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Course feedback

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