2. Making Mountains Out of Molehills The Banach-Tarski Paradox By Bob Kronberger Jay Laporte Paul Miller Brian Sikora Aaron Sinz

3. Introduction Definitions Schroder-Bernstein Theorem Axiom of Choice Conclusion

6. Banach-Tarski Theorem If X and Y are bounded subsets of having nonempty interiors, then there exists a natural number n and partitions and of X and Y (into n pieces each) such that is congruent to for all j.

7. Definitions Rigid Motions Partitions of Sets Hausdorff Paradox Piecewise Congruence

8. Rigid Motions

9. Rigid Motion

10. Partition of Sets A partition of a set X is a family of sets whose union is X and any two members of which are identical or disjoint.

11. Partition of Sets

12. Hausdorff Rotations

13. Hausdorff

14. Hausdorff Rotations

16. Piecewise Congruence

19. Schröder-Bernstein Theorem Theorem: If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|.

20. Cardinality Questions that need to be answered:  What is cardinality of sets?  How do you compare cardinalities of different sets?

21. Cardinality Definition:  Number of elements in a set.  Relationship between two cardinalities determined by: existence of an injection function existence of a bijection function

22. Cardinality Bijection function  One-to-one  Onto

23. Cardinality Bijection function  One-to-one  Onto Injection function  One-to-one

24. Cardinality

25. Comparing cardinalities of two finite sets  Both cardinalities are integers If integers are =  Bijection exists If integers are  No Bijection exists  Injection exists

26. Cardinality Comparing cardinalities of two infinite sets  Cardinality =  Cardinality

27. Cardinality Comparing cardinalities of two infinite sets  Cardinality =  Cardinality  Not always clear Z Bijection function 

28. Cardinality  Comparing cardinalities of a finite and an infinite  Infinite cardinality > finite cardinality

29. Schröder-Bernstein Theorem Four cases for sets A & B  Case I: A finite & B finite  Case II: A infinite & B infinite  Case III: A finite & B infinite  Case IV: A infinite & B finite Schröder-Bernstein Theorem: If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|

30. Schröder-Bernstein Theorem Four cases for sets A & B  Case I: A finite & B finite  Case II: A infinite & B infinite  Case III: A finite & B infinite  Case IV: A infinite & B finite Schröder-Bernstein Theorem: If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|

31. Schröder-Bernstein Theorem Two cases for sets A & B  Case I: A finite & B finite  Case II: A infinite & B infinite Schröder-Bernstein Theorem: If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|

32. Schröder-Bernstein Theorem Case I: A finite & B finite  |A| & |B| are integers  Let |A| = r, |B| = s Given conditions |A| ≤ |B| & |B| ≤ |A|, Given conditions r ≤ s & s ≤ r, then r = s |A| = |B| Schröder-Bernstein Theorem : If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|

33. Schröder-Bernstein Theorem Case II: A infinite & B infinite First condition Schröder-Bernstein Theorem:  If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| Injection function f from A into a subset of B,

34. Schröder-Bernstein Theorem Case II: A infinite & B infinite Second condition Schröder-Bernstein Theorem:  If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| Injection function g from B to a subset of A,

35. Case II: A infinite & B infinite Result Schröder-Bernstein Theorem:  If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| Bijection function h between A and B

36. Schröder-Bernstein Theorem Case II: A infinite & B infinite To get resulting bijection function h:  Combine the two given conditions Remove some of the mappings of g Reverse some of the mappings of g 

37. Schröder-Bernstein Theorem Resulting bijection function h |A| = |B|

38. The Axiom of Choice For every collection A of nonempty sets there is a function f such that, for every B in A, f(B) B. Such a function is called a choice function for A.

39. Galaxy O’ Shoes

40. Questions That Surround the Axiom of Choice 1. Can It Be Derived From Other Axioms? 2. Is It Consistent With Other Axioms? 3. Should We Accept It As an Axiom?

41. The First Six Axioms Axiom 1Two sets are equal if they contain the same members. Axiom 2For any two different objects a, b there exists the set {a,b} which contains just a and b. Axiom 3For a set s and a “definite” predicate P, there exists the set Sp which contains just those x in s which satisfy P. Axiom 4 For any set s, there exists the union of the members of s-that is, the set containing just the members of the members of s. Axiom 5For any set s, there exists the power set of s-that is, the set whose members are just all the subsets of s. Axiom 6There exists a set Z with the properties (a) is in Z and (b) if x is in Z, the {x} is in Z.

42. Can It Be Derived From Other Axioms?

43. Is It Consistent With Other Axioms?

44. Major schools of thought concerning the use of the Axiom of Choice A. Accept it as an axiom and use it without hesitation. B. Accept it as an axiom but use it only when you can not find a proof without it. C. Axiom of Choice is unacceptable.

45. Three major views are:  Platonism  Constructionism  Formalism

46. Platonism: A Platonist believes that mathematical objects exist independent of the human mind and a mathematical statement, such as the Axiom of Choice is objectively true or false.

47. Constructivism: A Constructivist believes that the only acceptable mathematical objects are ones that can be constructed by the human mind, and the only acceptable proofs are constructive proofs

48. Formalism: A Formalist believes that mathematics is strictly symbol manipulation and any consistent theory is reasonable to study.

49. Against:  Its not as simple, aesthetically pleasing, and intuitive as the other axioms.  With it you can derive non-intuitive results such as the Banach-Tarski Paradox.  It is nonconstructive

50. For:  Every vector space has a basis  Tricotomy of Cardinals: For any cardinals k and l, either k 1.  The union of countably many countable sets is countable.  Every infinite set has a denumerable subset.

51. What is a mathematical model?

52. What does the Banach-Tarski Paradox show?

53. Conclusion

54. References  Dr. Steve Deckelman  “The Banach-Tarski Paradox”  By Karl Stromberg  “The Axiom of Choice”  By Alex Lopez-Ortiz  “ Proof, Logic and Cojecture: The Mathematicians’”  By Robert S. Wolf