Re: aleph-null versus aleph-one

[John Denker <jsd@MONMOUTH.COM>]

> At 3:35 PM -0400 8/11/00, I wrote:
>
> >
> >The number of rationals (aleph-null) is indeed much less than the number of
> >reals ...

At 05:49 PM 8/14/00 -0600, Larry Smith wrote:

> What does "much" mean in this context?

Executive summary:
  In this case, "much less" means "infinitely less" :-)




============================================
If you want more quantitative detail:

  How many integers are there?  Aleph_null.
  How many rationals are there?  Aleph_null.
  Aleph_null - 20 = Aleph_null.
  Aleph_null / 20 = Aleph_null.

  2 ^ 20 is bigger than 20.
  2 ^ 30 is a lot bigger than 30.

  2 ^ Aleph_null = Aleph_one (definition of Aleph_one).
  Aleph_one is infinitely bigger than Aleph_null.
  Aleph_one - Aleph_null = Aleph_one
  Aleph_one / Aleph_null = Aleph_one

  The real line is sometimes called "the continuum".
  How many reals are there?  C.
  C is infinitely bigger than Aleph_null.
  C - Aleph_null = C.
  C / Aleph_null = C.
  C <= Aleph_one.

  You can assume C = Aleph_one if you like; many people do.
  This is called the continuum hypothesis.
  Nobody will ever disprove the continuum hypothesis.
  Nobody will ever prove it, either.
  It is provably not provable, just as in Euclidean geometry
    the parallel postulate is provably not provable.
  If you invoke the continuum hypothesis, it is good
    etiquette to say so explicitly.