Making Mountains Out of Molehills The Banach-Tarski Paradox

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

fixed. is 3

androtation fixed a is where3

for

)()( form thehaving 33

of mappingA

RaRx

axxrRRr

Rigid Motion

333231

232221

131211

1det T-1

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

X

jiXX

XXXX

ji

n

if

and

...21

Hausdorff Rotations

100

02123

02321

cos0sin

010

sin0cos

2

1sin

2

1cos xx

Hausdorff

31

221

321

23

22

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countable is

such that

subsetsfour into 1:

sphereunit theof ,,, partition a exists There

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xxxxRxS

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Hausdorff Rotations

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(1)

21

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n

i

nj

n

1

1

Hausdorff Rotations

22

m

m

m

m

PPP

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21

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Piecewise Congruence

YX

njXf

njf

njX

YX

jj

j

j

~

1:

1:

1:

~

Piecewise Congruence

YXXYYXvii

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symmetricXYYXii

reflexiveXXi

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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