Quote:
Originally Posted by ceptimus
Different sorts of infinity: This might be slightly heavy for a 'trivia' thread, but I'll try and keep it light. 
Infinity does tend to boggle the mind at first, but mathematicians encounter it pretty often and tend to become immune.  Cantor showed that there are different classes of infinity - in a loose sense, some infinities are 'bigger' than others.
Cantor called the 'smallest' infinity aleph-null - this is the infinity of the integers: 1, 2, 3, 4, ... but (and this may seem paradoxical) the same infinity holds for the primes: 2, 3, 5, 7, 11, 13, 17, ... or all the even integers, or all the numbers divisible by one-million, or all the rational fractions.
It seems strange at first - obviously there are twice as many integers as there are even integers, and there are a million integers for every one that is a multiple of a million - and there are a 'lot more' fractions than there are integers. How can we say that all these infinities are the same?
Georg Ferdinand Ludwig Philipp Cantor said that any set of things that could be put in a one-to-one correspondence with the counting numbers has the same 'cardinal' aleph-null. He then went on to use his famous 'diagonal' proof to show that not only are there are higher infinities than aleph-null, but there are an infinite number of them. 
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I love the infinites and transfinites. The
wiki on the alephs is rather well done.