Name: Anonymous 2007-08-17 21:40 ID:gpAqv6rZ
Not really. Time for some math.
So, this anon's been doing some reading about infinite sets. In particular, the Cantor set. The book I'm reading asserts (and proves - the "consider the numbers in the Cantor set written in ternary" proof, if you're someone who knows these things) that the Cantor set's cardinal number is the same as the real numbers'.
Now, the definition of the Cantor set suggests to me, a neophyte in these matters, that it should consist of rational numbers, in particular those between zero and one (inclusive) with denominator a power of three, and not even all of those. If it were only rationals, you'd expect it to have the same cardinality as the rationals, i.e. less than the reals. So, what's the deal here? Does the Cantor process describe real numbers in the limit kinda like describing real numbers by infinite nests of intervals a la Dedekind? Is there an alternative method of proof that makes it a little clearer how the definition of the Cantor set winds up spitting out real numbers as well as rationals?
So, this anon's been doing some reading about infinite sets. In particular, the Cantor set. The book I'm reading asserts (and proves - the "consider the numbers in the Cantor set written in ternary" proof, if you're someone who knows these things) that the Cantor set's cardinal number is the same as the real numbers'.
Now, the definition of the Cantor set suggests to me, a neophyte in these matters, that it should consist of rational numbers, in particular those between zero and one (inclusive) with denominator a power of three, and not even all of those. If it were only rationals, you'd expect it to have the same cardinality as the rationals, i.e. less than the reals. So, what's the deal here? Does the Cantor process describe real numbers in the limit kinda like describing real numbers by infinite nests of intervals a la Dedekind? Is there an alternative method of proof that makes it a little clearer how the definition of the Cantor set winds up spitting out real numbers as well as rationals?