Difference between revisions of "Banach-Tarski Paradox"

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The Banach-Tarski paradox is that statement that a ball can be cut into 5 pieces, which can then be reassembled into two balls of equal volume. It relies on the Axiom of Choice. It is called a paradox because it violates our physical intuition that this is impossible, though this paradox has been proven under Zermelo-Fraenkel set theory. Note that, since the pieces must be non-measurable, this cannot be done with an actual object.

The Mathematics Behind It

This section contains material appropriate for an college undergraduate.

Theorem

All sets discussed here are point sets in $\mathbb {R} ^{3}\$ .

A rigid motion of a set is a translation, rotation, or a composition of those two kinds of motion. A rigid motion of a set $A\$ is often written $g(A)\$ .

Let $A,B\$ be sets. We say there is a rigid body equivalence $A~B\$ if $A\$ can be partitioned into finitely many sets $A_{1},...,A_{n}\$ and there are $n\$ rigid motions $g_{1},...,g_{n}\$ such that $g_{1}(A_{1})\cup ...\cup g_{n}(A_{n})=B\$ .

We say there exists a paradoxical decomposition of a set $A\$ if $A=B\cup C\$ , and $A~B\$ and $A~C\$ .

The Banach-Tarski theorem (aka the Banach-Tarski paradox) states that the solid ball admits a paradoxical decomposition.

Removing Points with Rigid Body Motions

The Free Group

Paradoxical Decomposition of the Free Group

A Geometric Model of the Free Group

The Axiom of Choice

The Banach-Tarski Construction

Interpretation

Some view the Banach-Tarski paradox as an absurd result, and evidence that the Axiom of Choice is false. This is the reaction of many non-mathematicians upon hearing this result. However, as one grows in mathematical understanding one finds it is impossible to avoid - mathematicians have learned to accept the existence of non-measurable sets, comfortable in the knowledge that they can never exist in the real world.

It can also be viewed as a mathematical affirmation of the biblical story of the loaves and the fishes (as Jesus would not have had to violate the conservation of mass, but merely partition loaves and fishes into non-measurable subsets).

External Links

Banach-Tarski Paradox at MathWorld

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 {{disputed}}
-The '''Banach-Tarski paradox''' uses the [[Axiom of Choice]] to take a [[sphere]], split it into finitely many pieces, and reassemble them into two new spheres. The new spheres both have the same volume as the original, suggesting paradoxically that you can create two identical copies out of one, despite our physical intuition that this is impossible. This paradox has been mathematically proven to be consistent with [[Zermelo-Fraenkel]] set theory.
+The '''Banach-Tarski paradox''' is that statement that a ball can be cut into 5 pieces, which can then be reassembled into two balls of equal volume.  It relies on the [[Axiom of Choice]].   It is called a paradox because it violates our physical intuition that this is impossible, though this paradox has been proven under [[Zermelo-Fraenkel]] set theory.
 Note that, since the pieces must be non-measurable, this cannot be done with an actual object.
+==The Mathematics Behind It==
+:''This section contains material appropriate for an college undergraduate.''
+===Theorem===
+All sets discussed here are point sets in <math>\mathbb{R}^3 \ </math>.
+A '''rigid motion''' of a set is a translation, rotation, or a composition of those two kinds of motion.  A rigid motion of a set <math>A \ </math> is often written <math>g(A) \ </math>.
+Let <math>A, B \ </math> be sets.  We say there is a '''rigid body equivalence''' <math>A~B \ </math> if <math>A \ </math> can be partitioned into finitely many sets <math>A_1, ..., A_n \ </math> and there are <math>n \ </math> rigid motions <math>g_1, ..., g_n \ </math> such that <math>g_1(A_1) \cup ... \cup g_n(A_n) = B \ </math>.
+We say there exists a '''paradoxical decomposition''' of a set <math>A \ </math> if <math>A = B \cup C \ </math>, and <math>A~B \ </math> and <math>A~C \ </math>.
+The Banach-Tarski theorem (aka the Banach-Tarski paradox) states that the solid ball admits a paradoxical decomposition.
+===Removing Points with Rigid Body Motions===
+===The Free Group===
+===Paradoxical Decomposition of the Free Group===
+===A Geometric Model of the Free Group===
+===The Axiom of Choice===
+===The Banach-Tarski Construction===
 ==Interpretation==
-Some view the Banach-Tarski paradox as an absurd result, and evidence that the [[Axiom of Choice]] is false. However, it can also be viewed as a mathematical affirmation of the biblical story of the loaves and the fishes (as Jesus would not have had to violate the conservation of mass, but merely partition loaves and fishes into non-measurable subsets).
+Some view the Banach-Tarski paradox as an absurd result, and evidence that the [[Axiom of Choice]] is false.  This is the reaction of many non-mathematicians upon hearing this result.  However, as one grows in mathematical understanding one finds it is impossible to avoid - mathematicians have learned to accept the existence of non-measurable sets, comfortable in the knowledge that they can never exist in the real world.
+It can also be viewed as a mathematical affirmation of the biblical story of the loaves and the fishes (as Jesus would not have had to violate the conservation of mass, but merely partition loaves and fishes into non-measurable subsets).
 ==External Links==