
Counting Infinity - soundsop
http://duartes.org/gustavo/blog/post/counting-infinity
======
newt0311
Infinity is one of those ideas which is a heavily researched and interesting
idea in mathematics which has very little applicability elsewhere. Check out
an intro analasys book for more details on countable and uncountable sets. The
cantor set especially, is very interesting as a set with measure 0 and still
uncountable many elements.

PS. the use of aleph-0 for countable sets is discuraged as we do not know is
there exists a set with cardinality between the countable and uncountable sets
(ie. a set which does not have a bijection with the natural numbers but does
have a bijection with a "small" subset of the reals such that it did not have
a bijection with the reals themselves. ). Its called the continuum hypothesis.
Details here: <http://en.wikipedia.org/wiki/Continuum_hypothesis>

~~~
gjm11
The use of aleph-0 for countable sets is not discouraged. I think what you
mean is that using aleph-1 to mean "of the same cardinality as the reals" is
discouraged, because it's only appropriate in models where the continuum
hypothesis holds.

There's a not-so-commonly-used notation using the Hebrew letter beth instead
of alpha, where beth_0 = aleph_0 and beth_{n+1} = 2^beth_n (note for experts:
and you do the obvious thing at limit ordinals). So the cardinality of the
real numbers is beth-1, whatever model you're in.

