Making Mountains Out of Molehills The Banach-Tarski Paradox

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Making Mountains Out of MolehillsThe
Banach-Tarski Paradox

• By
• Bob Kronberger
• Jay Laporte
• Paul Miller
• Brian Sikora
• Aaron Sinz

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Introduction
Definitions Schroder-Bernstein Theorem Axiom of
Choice Conclusion
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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.

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Definitions

• Rigid Motions
• Partitions of Sets
• Hausdorff Paradox
• Piecewise Congruence

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Rigid Motions
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Rigid Motion
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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.

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Partition of Sets
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Hausdorff Rotations
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Hausdorff
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Hausdorff Rotations
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Hausdorff Rotations
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Piecewise Congruence
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Piecewise Congruence
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Piecewise Congruence

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Schröder-Bernstein Theorem

• Theorem
• If A B and B A, then A B.

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Cardinality

• Questions that need to be answered
• What is cardinality of sets?
• How do you compare cardinalities of different
  sets?

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Cardinality

• Definition
• Number of elements in a set.
• Relationship between two cardinalities
  determined by
• existence of an injection function
• existence of a bijection function

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Cardinality

• Bijection function
• One-to-one
• Onto

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Cardinality

• Bijection function
• One-to-one
• Onto
• Injection function
• One-to-one

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Cardinality

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Cardinality

• Comparing cardinalities of two finite sets
• Both cardinalities are integers
• If integers are
• Bijection exists
• If integers are
• No Bijection exists
• Injection exists

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Cardinality

• Comparing cardinalities of two infinite sets
• Cardinality
• Cardinality

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Cardinality

• Comparing cardinalities of two infinite sets
• Cardinality
• Cardinality
• Not always clear
• Z
• Z
• Bijection function

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Cardinality

• Comparing cardinalities of a finite and an
  infinite
• Infinite cardinality gt finite cardinality

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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

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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

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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

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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

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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,

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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,

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• 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

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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

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Schröder-Bernstein Theorem

• Resulting bijection function h
• A B

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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.

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Galaxy O Shoes
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Questions That Surround the Axiom of Choice

• Can It Be Derived From Other Axioms?
• Is It Consistent With Other Axioms?
• Should We Accept It As an Axiom?

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The First Six Axioms

• Axiom 1 Two sets are equal if they contain the
  same members.
• Axiom 2 For any two different objects a, b there
  exists the set a,b which contains just
  a and b.
• Axiom 3 For 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 5 For any set s, there exists the power set
  of s-that is, the set whose members are just
  all the subsets of s.
• Axiom 6 There exists a set Z with the properties
  (a) is in Z and (b) if x is in
  Z, the x is in Z.

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Can It Be Derived From Other Axioms?
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Is It Consistent With Other Axioms?
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Major schools of thought concerning the use of
the Axiom of Choice

• Accept it as an axiom and use it without
  hesitation.
• Accept it as an axiom but use it only when you
  can not find a proof without it.
• Axiom of Choice is unacceptable.

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Three major views are

• Platonism
• Constructionism
• Formalism

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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.

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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

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Formalism

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

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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

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For

• Every vector space has a basis
• Tricotomy of Cardinals For any cardinals k and
  l, either klt1 or k1 or kgt1.
• The union of countably many countable sets is
  countable.
• Every infinite set has a denumerable subset.

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What is a mathematical model?
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What does the Banach-Tarski Paradox show?
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Conclusion
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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