From the Quaternions Sent Fri, Jun 11th 1999, 03:30
From the Quaternions Sent Fri, Jun 11th 1999, 03:30
> There is, in fact, a hierarchy of infinities. >> > > How do we know that this is true? > > I'm not necessarily disputing it, just want to know what the proof is. One way of thinking about it has to do with matching. If you can pair any given element in one infinity up with any given element in another, they're the same "cardinality." So, for example, the integers are the same cardinality as the even numbers, because x can always be paired with 2x. But, for things like the real numbers (i.e. any number that can be represented as a fraction), you can prove that there is no one-to-one pairing possible. Thus, reals have a higher cardinality than integers... Or, think of it this way. You can count integers 1-2-3-4... But how do you "count" fractions? There's always an infinite number of fractions between any two given fractions. It's just like there are an infinite number of songs you can write with a 256 tone keyboard, but the infinity is of another quality entirely when you're talking about infinitely nuance-able physical instruments, or electronic gear that can be tweaked to any specification. The first has infinity of melody. The second, infinity of harmony and infinity of tone--a double infinity, if you will. Sorry for the non-IDM nature of the post. Sam. High school math nerd.