4 ms·

More troubling to me has been the continuum hypothesis. The notion that you can toss a dart (whose tip is the width of a point) at the real number line and hit

More troubling to me has been the continuum hypothesis. The notion that you can toss a dart (whose tip is the width of a point) at the real number line and hit an integer or rational number with probability zero is very unintuitive. I can kind of grasp the continuum hypothesis, there does seem to be a distinction between countable and uncountable infinity. Continuing the game beyond that to this area of large cardinals strikes me as just a language game. Most mind blowing, IMHO, is Cantor's middle third's set[1]. Uncountable nowhere dense totally disconnected, ... it's an amazing set and so easy to construct. [1] http://en.wikipedia.org/wiki/Cantor_set http://en.wikipedia.org/wiki/Cantor_set

stanleydrew 17y ago
You're not talking about the continuum hypothesis. The continuum hypothesis just states that there isn't any set with cardinality larger than the natural numbers (aleph_0) but smaller than the reals (2^aleph_0). Interestingly this can't be proven or disproven within set theory if you assume the axiom of choice. Comparing the cardinality of the natural and real numbers doesn't have much to do with the continuum hypothesis. It's simply true that there are many many many more reals than rationals. It's not a hypothesis. But I do agree that the Cantor set is pretty awesome.
- bitdiddle 17y ago
  Actually I was talking about it. My mind wanders. Sorry to be so vague (it's been years since I read this stuff). I didn't mean to imply sets of measure zero were what CH is about. The reals can be essentially equated with the powerset of the naturals. It's very intriguing that there is no set with cardinality in between that of the naturals and that of the reals. The Axiom of Choice is also independent of ZF set theory. Have you read the more modern expositions of these independence proofs based on various Topoi?

stanleydrew 17y ago

You're not talking about the continuum hypothesis. The continuum hypothesis just states that there isn't any set with cardinality larger than the natural numbers (aleph_0) but smaller than the reals (2^aleph_0). Interestingly this can't be proven or disproven within set theory if you assume the axiom of choice. Comparing the cardinality of the natural and real numbers doesn't have much to do with the continuum hypothesis. It's simply true that there are many many many more reals than rationals. It's not a hypothesis. But I do agree that the Cantor set is pretty awesome.

bitdiddle 17y ago
Actually I was talking about it. My mind wanders. Sorry to be so vague (it's been years since I read this stuff). I didn't mean to imply sets of measure zero were what CH is about. The reals can be essentially equated with the powerset of the naturals. It's very intriguing that there is no set with cardinality in between that of the naturals and that of the reals. The Axiom of Choice is also independent of ZF set theory. Have you read the more modern expositions of these independence proofs based on various Topoi?