On Sun, May 29, 2011 at 9:56 AM, Nicholas Leippe <[email protected]> wrote:
>> They all have the same cardinality as the natural numbers in general. >> You can make a list of the first prime number, the second prime number, >> etc. This infinite list is a one-to-one and onto function from the >> natural numbers to the prime numbers (proof not included in this email), >> so the two sets have the same cardinality. > > That is extremely counter-intuitive to me. The way I've been looking > at it is by starting with the set of all natural numbers, yielding > cardinality X, then removing from that set any number that doesn't > belong in the other set (such as not prime, or not even, or not > fibonacci) until the resultant set is achieved--and it *seems* like > that would be fewer numbers and thus the cardinality would be less > than X. > > Just taking the set of even numbers, for example, why isn't the > cardinality of that set 1/2 the cardinality of all natural numbers? I think you'll find Hilbert's paradox of the Grand Hotel interesting, as it deals with this particular odd property of infinite sets: http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel This whole issue with the counterintuitive bits of mathematics is part of why the cosmology discussion was so difficult. Most of modern physics is highly mathematical, and a lot of that math is counterintuitive despite yielding results that match up very well with experimental data. Because the math is counterintuitive, it's difficult to translate into English in a way that both makes sense and corresponds to the math. --Levi /* PLUG: http://plug.org, #utah on irc.freenode.net Unsubscribe: http://plug.org/mailman/options/plug Don't fear the penguin. */
