Re:Donald Knuth's letter about teaching O notation in calcul...
by yters on Monday April 21, 2008 @12:34AM
(#169076)
You don't think mathematics is unified, i.e. something true in one area may not be true in another? I don't think a lot of mathematicians believe this, and it isn't clear to me that "infinity" refers to two different things in the two different realms (the other possibility).
So, in set theory, cardinality is a measure of the elements of a set. The cardinality of the infinite set of natural numbers is aleph-0. An infinite set has cardinality aleph-0 if it can be put in one to one correspondence with the naturals. http://en.wikipedia.org/wiki/Bijection This counterintuitively includes the rational numbers also. http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
The real numbers can't be put into one-to-one correspondence with the rational numbers. So the infinite of the reals truly is bigger than the infinite of the rational numbers.
Also see this ppt: http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/di...