God and Math

A while ago I challenged a friend to a math duel. The rules: we send original proofs back and forth, each trying to refute the other's, and at the end of the school year present them to our former math teacher, who will judge which is the 'coolest' proof and also the 'most impressive'. For the opening proof I went big:

Prove Euclid's definition of a point: 'that which has no part'.

I'll save you the infinite series notations, and explain this in a pseudo-philosophic way. Take a line segment, from 0 to 1. Now remove the middle third, inclusively, such that you have two line segments, from 0 to 1/3, and 2/3 to 1, including all end points. Now remove the middle third of those two, and keep doing that infinitely. Lets define the term "length" to mean the total "unremoved" portions, so at the beginning it is 1, then 2/3, and next 4/9. (This is called the 'Cantor Set', after a certain Georg Cantor, who did a lot of work with infinity, and such). The length becomes infinitely smaller, until it reaches 0 (this can be proven with an infinite series). Yet there are also an infinite number of points, since the end points can never be in the middle third, thus 0, 1/3, 2/3, and 1 are four of the infinitely many end points. This means that there is infinitely many points within 0 and 1, but infinitely little length. Therefore, a point has no part.

I thought that was cool enough, but then I encountered another Greek, Zeno of Elea. He cleverly pointed out that if a point has no part, no matter how many of them you stick together you will never get anything of substance (a line). So in the first place we have proven mathematically that a point does not have any length, yet that means that infinitely many points of no length somehow constitute length.

And people say there is not a God?