Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?]

[ikastan@alumnae.caltech.edu (Ilias Kastanas)]

In article <JSA.96Sep27200651@alexandria>,
Jon S Anthony <jsa@alexandria> wrote:
>
>In article <52eha1$o7h@krusty.irvine.com> adam@irvine.com (Adam Beneschan) writes:

...

>> However, no one told me whether it was known whether C=Aleph1 or not
>> in the "usual case" of the real numbers and integers.
>
>That's probably because it is not known :-), and may be unknowable in
>any meaningful sense.


	It is known, in a sense, about definable sets of reals:  Cantor
   showed that any closed set is either countable or it has the cardinality
   of the reals.  This can be extended to all Borel sets, and in fact to
   analytic sets (continuous images of Borel sets).  At that point, however,
   independence sets in.  One can proceed by complements and continuous
   images to define higher and higher levels of "projective" sets... but one
   needs more than classical set theory to handle them.  If one adds the
   axiom of Projective Determinacy, then all projective sets satisfy CH,
   like the analytic ones.  Of course, there are lots of _other_ sets of
   reals...

							Ilias