Tuesday, April 30, 2013

To infinity and beyond!

When you have an opportunity, ask a child if he wants to win one or two candies.*
If the child is rational, it seems quite obvious that he will take 2. "More is better", he would think.

This kind of reasoning doesn't hold in a special world; the magical world of infinites.
When we come into this subject any basic arithmetic is useless.

Georg Cantor left many insights to humankind about the conundrums of infinite. He once said, shocked with his own discoveries: "I see it, but I do not believe it". And unsatisfied with a lack of understanding of infinite, he came with the idea that there is an infinity of infinites. Yes, you read it right, an infinity of infinites.

In order to give you a simple example of how our basic arithmetic doesn't work in this strange world, let's take two sets:

E = { 2, 4, 6, ... }
N = { 1, 2, 3, 4, 5, 6, ... }

Which one has more elements, N or E?

The common reasoning is to think that E has fewer elements than N because E is a subset of N; a valid thought if they were finite sets. But here, our intuition doesn't suit. 

N and E, surprisingly, have the same number of elements! In other words, the number of natural numbers is equal to the number of even numbers!
But, but...how can it be?!

The key to understand this idea is one-to-one correspondence. An even number always has a half representation in the natural numbers. If we map all the even numbers to their half representations, we can check that for each element in set E there is an element in the set N. Hence, they have the same number of elements, or technically speaking, the same cardinality.

Strikingly simple as that, and at the same time mysterious. 

Despite all Cantor's et al endeavours, we still have a lot more to learn from 

Till the next post!
Ronald Kaiser

* Please, don't do that if you do not have 2 candies in hand, =P