Tuesday, November 21, 2006


I watched a history channel special the other night on UFOs. It had some interesting interviews on it. One thing that was eye opening was the replay of a nasa astronaut saying over the radio that he was looking at the alien spaceship out of his window. He said it very matter-of-factly and very clearly. He was neither surprised nor concerned. I have no problem believing in aliens. However, the Roswell stuff is a bit far fetched. How is it that the aliens can fly billions of miles across the universe and then crash in New Mexico? Doesn't make a whole lot of sense to me.

However, I propose the following. I think the next candidate for president of the United States should run on a platform of revealing all government knowledge of UFOs. I'd vote for that person, I think it'd be cool to know. What about you?

Monday, November 20, 2006

Unfair Infinities

We all know that the cardinality of the set of all integers is countably infinite. To me, it seems unfair that the cardinality of the set of real numbers is uncoutably infinite. It also seems odd that it is unintuitive to most (including me). However, I now understand why it was unintuitive to me. The key point to remember is that mathematicians don't allow integers to have an infinite number of digits. Instead, you can increase the integer by 1, but at any given time it has a fixed number of digits. However, with real numbers, there can, and often does, exist an infinity of digits. For instance, 1/3 is 0.333333.... So, not only do the real numbers extend countably to infinity, they also can have an infinite number of digits. In essence, they are infinite in 2 dimensions, whereas integers are only infinite in 1 dimension. I would imagine that complex numbers will one day be proven to be infinite in 3 dimensions, but who knows.

To be honest, I think it is a bit silly. We're not actually talking about different orders of infinities here, we're only talking about the existence of a function M that will map from one infinite set to another. If we can find that function, then we say the sets have an equivalent cardinality. Otherwise, the cardinality of one set is said to be greater than the cardinality of the other. If we could extract M from mathematics, and instead use an algorithm, then M could simply be, pick a real number, get next biggest integer, rinse, repeat. However, algorithms are not yet mathematical (unless you're Stephen Wolfram).