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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\cong 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = B \cup C \ } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\cong B \ } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\cong 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

@@ Line 13: / Line 13: @@
 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>.
+Let <math>A, B \ </math> be sets.  We say there is a '''rigid body equivalence''' <math>A \cong 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>.
+We say there exists a '''paradoxical decomposition''' of a set <math>A \ </math> if <math>A = B \cup C \ </math>, and <math>A\cong B \ </math> and <math>A\cong C \ </math>.
 The Banach-Tarski theorem (aka the Banach-Tarski paradox) states that the solid ball admits a paradoxical decomposition.