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Making Mountains Out of Molehills The Banach-Tarski Paradox. By Bob Kronberger Jay Laporte Paul Miller Brian Sikora Aaron Sinz. Introduction. Definitions Schroder-Bernstein Theorem Axiom of Choice Conclusion. Banach-Tarski Theorem. - PowerPoint PPT Presentation
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Making Mountains Out of MolehillsThe Banach-Tarski Paradox
By
Bob Kronberger
Jay Laporte
Paul Miller
Brian Sikora
Aaron Sinz
Introduction
Definitions
Schroder-Bernstein Theorem
Axiom of Choice
Conclusion
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.
3R
njX j 1: njY j 1:
jX
jY
Definitions
Rigid Motions
Partitions of Sets
Hausdorff Paradox
Piecewise Congruence
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.
Partition of Sets
thatmeans subsets into ofpartition a is }1:{ nXnjj
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and
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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
YXXYandYXvii ~~~
Schröder-Bernstein Theorem
Theorem:
If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|.
Cardinality
Questions that need to be answered:What is cardinality of sets?How do you compare cardinalities of different sets?
Cardinality
Definition: Number of elements in a set. Relationship between two cardinalities determined by:
•existence of an injection function•existence of a bijection function
Cardinality
Bijection functionOne-to-oneOnto
Cardinality
Bijection functionOne-to-oneOnto
Injection functionOne-to-one
Cardinality
B? &A between function bijection
B intoA fromfunction injection |B| |A| If
B intoA fromfunction injection |B| |A| If
B &A between function bijection |B| |A| If
Cardinality
Comparing cardinalities of two finite setsBoth cardinalities are integers
• If integers are = Bijection exists
• If integers are No Bijection exists Injection exists
Cardinality
Comparing cardinalities of two infinite setsCardinality =Cardinality
Cardinality
Comparing cardinalities of two infinite setsCardinality =CardinalityNot always clear
• Z • Z• Bijection function
2
2
xxf 2
Cardinality
Comparing cardinalities of a finite and an infinite Infinite cardinality > finite cardinality
Schröder-Bernstein Theorem
Four cases for sets A & BCase I: A finite & B finiteCase II: A infinite & B infiniteCase III: A finite & B infiniteCase IV: A infinite & B finite
Schröder-Bernstein Theorem:If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|
Schröder-Bernstein Theorem
Four cases for sets A & BCase I: A finite & B finiteCase II: A infinite & B infiniteCase III: A finite & B infiniteCase IV: A infinite & B finite
Schröder-Bernstein Theorem:If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|
Schröder-Bernstein Theorem
Two cases for sets A & BCase I: A finite & B finiteCase II: A infinite & B infinite
Schröder-Bernstein Theorem:If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|
Schröder-Bernstein Theorem
Case I: A finite & B finite |A| & |B| are integersLet |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|
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, B
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, A
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
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
-1g
Schröder-Bernstein Theorem
Resulting bijection function h |A| = |B|
Af\Bg\A
Af\B1
xxf
xxgxh
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.
Galaxy O’ Shoes
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?
The First Six AxiomsAxiom 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.
Can It Be Derived From Other Axioms?
Is It Consistent With Other Axioms?
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.
Three major views are:
PlatonismConstructionismFormalism
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.
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
Formalism:A Formalist believes that mathematics
is strictly symbol manipulation and any consistent theory is reasonable to study.
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
For:Every vector space has a basisTricotomy of Cardinals: For any cardinals k and l, either k<1 or k=1 or k>1.The union of countably many countable sets is countable.Every infinite set has a denumerable subset.
What is a mathematical model?
What does the Banach-Tarski Paradox show?
?
Conclusion
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