by Chris Bodenner
A reader writes:
You think that's bad? You would think that infinity is just infinity, but you would be wrong. There are in fact different sizes of infinity – infinitely many, in fact. This was proven by Georg Cantor in the late 19th century, using one of the most beautiful and simple proofs in the history of mathematics (now known as Cantor diagonalization). When he published his results, he was viciously attacked on both religious and philosophical grounds, his opponents charging that his theorems called into question the existence of God.
Another needs a translator:
Hilbert's example only uses countable infinity, or infinity where there is a one-to-one correspondence with whole numbers, or positive Integers. This is only the first order of magnitude (cardinality) of Infinite Sets, dubbed Aleph-Null (ran outta Greek letters and went to Hebrew). Next order of magnitude would be Aleph-one – the set of Real Numbers for example (provable depending on the Axiom of Choice and other stuff I don't quite remember), which includes the irrational numbers like pi, e, SQRT (2), etc. Go here, here and here for some fun (well, for some of us, anyway) reads. A bit easier to figger out: Zeno's paradox, explainable with calculus (convergent infinite series).
As it happens, the book I'm reading is Douglas Hofstadter's Gödel, Escher, Bach, which invokes Zeno's paradox in early chapters (and explains other complex concepts in a way that won't hurt your brain). A concise illustration from the reader's link:
Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.
As a little kid I had a similar thought while looking at my index fingers: they can never truly touch, since no matter how much closer they get to one another there will always be an infinitely small distance to go between them.
(Above is a live-action version of M.C. Escher's "Waterfall". Behind-the-scenes explanations here.)
The Dish 