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 a 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 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 \ } 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 $\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 $R_{\alpha }\$ on the necklace, we simply get $\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.

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Removing countably many points from a circle is a trickier matter. Let $A=\left\{{a_{1},a_{2},...}\right\}\$ be countable set of points in the circle, where it is understood that the $\alpha \$ measure the degree of the angle the point can be found at. We construct a set $W=\left\{{{\frac {a_{m}-a_{k}}{n}}:(m,k,n)\in \mathbb {N} ^{3}}\right\}\$, 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 $W\$. If the first few values of $A=\left\{{6,2{\sqrt {3}},{\frac {e}{\pi }},...}\right\}\$, then $W\$ contains, among other things, $6-2{\sqrt {3}},(2{\sqrt {3}}-{\frac {e}{\pi }})/653,\$ etc. Now, since $W\$ is countable, and the angles of $\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.

Note that if we rotate 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 \ } by an angle 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 \ } , it is guaranteed to not carry any point 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 A \ } to any other point 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 A \ } because of our clever definition. Rotation by 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 \ } , for example, cannot carry 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_7 \ } to 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_11 \ } , because 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 \theta = a_{11}-a_7 \in W \ } , but we specifically said 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 \notin W \ } . Furthermore, 12 rotations 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 \theta \ } could never carry 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_7 \ } to 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_{11} \ } , because 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 \theta = \frac{a_{11}-a_7}{12} \ } , but that's 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 \ } too, since we constructed it to contain not just all the differences of angles 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 A \ } , but all the fractions of those differences as well.

So we construct the infinite super-necklace 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 T = A \cup R_\theta(A) \cup R_{2\theta}(A) ... \ } and finally, observe that 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_\theta(T) = R_\theta(A) \cup R_{2\theta}(A) \cup ... = T-A \ } .

We form the partition 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 = T \cup (\mathbb{S}^1-T) \ } and then note 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_\theta(T) \cup (\mathbb{S}^1 - T) = \mathbb{S}^1 - A \ } and we have shown that 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 \cong \mathbb{S}^1 - A \ }

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Removing countably many points from a sphere is similar, except now we are using the longitude of the point. A tricky technical matter arises:

The Free Group

Before attempting to understand the free group, it would be very beneficial to the reader to have and understanding of a mathematical group.

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,a',b,b' \ } be "letters." We will form "words" by stringing finitely many of these letters together, subject to just a few simplification rules: 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 aa' = a'a = bb' = b'b= e \ } , where 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 e \ } is the identity, shorthand for the "word" of zero letters.

We state without proof that such words form a group under "concatenation," or stringing back to back, ie, 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 aab' * bab' = aab'bab' = aaab' \ } . By using exponents, ie, 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 aaa = a^3 \ } , each word has a unique spelling.

This is called 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 F_2 \ } , the free group on two generators. Any group structured like this, in which a and b have infinite order and are independent, (ie, no combination of a and b form e) is this group. We can picture this group graphically, with a and b axes:

This depiction demonstrates words of up to four letters; a graphical depiction is fractal in a fairly obvious fashion.

Paradoxical Decomposition of the Free Group

U,L, and D are within the ovals. R is what is left over.

Why are we talking about the free group? It is important for two reasons, the first of which is that it admits a "paradoxical decomposition", if we allow such a notion to refer to the graphical representation of it above. We begin by partioning the group as 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 F_2 = U \cup D \cup L \cup R \ } , where the sets 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 U,D,L,R \ } are those described in the image to the right.

Consider 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 b'(U) = \left\{{b'u : u \in U }\right\} \ } . Since every element of U starts with 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 b^k \ } , followed by a letter 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, a' \ } , and then some other letters, every element 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 b'(U) \ } starts with 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 b, a, \ } or 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' \ } - that is, three of the four trees 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 F_2 \ } ! The only thing missing is 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 D \ } . Thus, 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 b'(U) \cup D = F_2 \ } . Likewise, 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'(R) \cup L = F_2 \ } . Hence, we have something akin to a "paradoxical decomposition" of the free group.

A Geometric Model of the Free Group

You may be unimpressed by this, since the free group isn't really a geometric object. If you recall, we said earlier that the free group was important for two reasons, and one of them was that it had a paradoxical decomposition. We now discuss the second reason.

Consider the unit sphere in xyz space. Pick a point on the sphere P, and 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 \ } mean rotation of this point counterclockwise around the x-axis by 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 \ } , 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 b \ } mean rotation around the z-axis counterclockwise by that same angle 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 \ } . If we define 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 \ } so that 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 \cos(\theta) = 1/3 \ } , it turns out that this is precisely the free group 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 F_2 \ } if we define 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 ab \ } to mean first doing one rotation, then the other, 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',b' \ } to be clockwise rotations respectively.

The isomorphism between this group, with 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 = \arccos(1/3) \ } and so on, is not easy, and requires some hard linear algebra. It will not be given here.

From now on, we will use 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{F} \ } to refer to this group of rotations on the sphere.

The Axiom of Choice

Before continuing to a complete proof, we must discuss the Axiom of Choice. The axiom states than given any collection of sets (possibly infinitely many sets - possibly uncountably many sets!), there exists a set which contains exactly one element of each.

The Banach-Tarski Construction

We begin by removing the center of the unit ball by choosing some circle in the ball which contains the center, and using the method by which we removed a point from the circle earlier. Now that we have done this, we can work on the surface of the ball, 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}^2 \ } , by understanding that everything we do on the surface can extend downward to the interior of the ball in a ray towards the center.

Now we define the orbit of a point x of the sphere under the operations of the free group to be 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{F}(x) \ } . For almost all the points on the surface, this will yield a copy of the free group, except a couple of bad points: for example, the "north pole" is fixed under the b operation. So define 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 B=\left\{{x\in\mathbb{S}^2: \exists r\in\mathbb{F}, (r\neq e) \and (r(x)=x) }\right\} \ } , that is, the set of all the points on the sphere which are held still by some rotation 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 F_2 \ } .

Since B is countable, and we can remove any countable set of points from the sphere, remove this 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 \mathbb{S}^2 - B \ } .

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.

External Links

• Banach-Tarski Paradox at MathWorld