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\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 $A=B\cup C\$, and $A\cong B\$ and $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

First, lets examine why a circle is rigid-body equivalent to a circle with a point removed. Pick an irrational number $\alpha \$, and let $R_{\alpha }\$ be the rigid motion which is rotation by $\alpha \$ degrees counterclockwise. Now we pick a point in the circle $P\$ and we form an infinite "necklace" $\left\{{P,R_{\alpha }(P),R_{2\alpha }(P),...}\right\}\$. It is clear that this necklace must be infinite, because if there was any $n\$ such that $R_{n\alpha }(P)=P\$, then 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 \alpha \ } would not be irrational.

Since our necklace is a countable set and the circle is uncountable, there are points in the circle which are not in the necklace. So we partition the circle into two sets, the necklace, and everything left over. Observe that by performing the rigid motion 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 R_\alpha \ } on the necklace, we simply get 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 \left\{{R_\alpha(P), R_{2\alpha}(P), ... }\right\} \ } , that is, all the original points remain exactly where they were, except now the first point is gone. Re-combining the rotated necklace with those points not in the necklace gives us a circle with a point removed.

With this procedure, it becomes a simple matter to remove a point from the sphere or the interior of a ball - just pick a circle lying in the sphere or ball which contains the point you wish to remove, and partition the sphere or ball into three pieces - the necklace, the rest of the circle the necklace is on, and the everything in the sphere or ball which isn't in that circle.

Removing countably many points from a circle is a trickier matter. Let 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 = \left\{{ a_1, a_2, ... }\right\} \ } be countable set of points in the circle, where it is understood that the 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 \alpha \ } measure the degree of the angle the point can be found at. We construct a set 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 W = \left\{{\frac{a_m-a_k}{n}:(m,k,n)\in \mathbb{N}^3 \ } , which is also a countable set, since it is in one-to-one correspondence with the three dimensional lattice.

Make sure you understand the content of 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 W \ } . If the first few values of 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 = \left\{{6,2\sqrt{3},\frac{e}{\pi}, ... }\right\} \ } , then 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 W \ } contains, among other things, 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 6-2\sqrt{3}, (2\sqrt{3}-\frac{e}{\pi})/653, \ } etc. Now, since 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 W \ } is countable, and the angles of 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{S}^1 \ } are uncountable, there is an angle 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{S}^1 \ } which is not 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 W \ } . Pick one such angle, and call it 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 \theta \ } . This is our magic rotation angle.

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