I left off last time explaining how invertible function spaces naturally form a group and how that leads to the concept of a group action.  This concept turns out to be useful in examining any group, not just explicit function spaces.  Any group G can be seen as a space of invertible functions (by the way, the fancy term for these sorts of function is bijections) on itself.  How does this work?  Well, let g be some element of G.  With g we can associate a function g(h) G->G such that g(h) = gh.  All this function does is multiply it's argument by g on the left using the normal group multiplication;  since g has an inverse in G, the inverse function is also well defined.  Now since we've presented the group as a function space we can define an action p of G on G using the evaluation function by saying that p(g, h) = gh. 

One more piece of notation.  Let's say that G acts on X and g is a group element.  If A is a subset of X, we define gA to be the set obtained by applying the action of g to every element in A.  In formal notation:  gA = {ga : a :el: A}

Now that group actions are understood, we can move on to a concept at the heart of the Banach Tarski Paradox.  Let's start with a definition in Math Speak:

   Let G be a group acting on a set X.  We say that this group action is paradoxical if there are disjoint subsets of X A1, A2, ..., An and B1, B2, ..., Bn and group elements g1, g2, ..., gn such that the following holds:

• g1A1 U g2 A2 U ... U gn An = X
• g1B1 U g2 B2 U ... U gn Bn = X

Less formally, this set X  can be pulled apart into two pieces, and each of these pieces can be grown back into the original set by proper application of group elements.

In and of itself, this result might not be terribly surprising.  After all, the function in G might be doing terribly strange things to the set X.  In fact if we let G be the group of all invertible functions on X, you can show right away that the group action is paradoxical.  What would be surprising is if we could show that an ordinary, familiar group and an ordinary, familiar action has this property.

And of course we can.  Let SO^3 be the group of all 3-dimensional rotations. The Banach Tarski Paradox states that the group action of SO^3 on a 3-dimensional ball with radius 1 is paradoxical.  Namely, we can pull the ball into several pieces, apply rotations to those pieces, and get two complete balls when we put them together.  What's more, this can be done using no more than 5 pieces.

Next time I'll outline the proof, and talk about how this impacts the real world.