From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-1.3 required=5.0 tests=BAYES_00,INVALID_MSGID autolearn=no autolearn_force=no version=3.4.4 X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: fc89c,97188312486d4578 X-Google-Attributes: gidfc89c,public X-Google-Thread: 109d8a,ae69c50ef02cd1c0 X-Google-Attributes: gid109d8a,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public X-Google-Thread: 109fba,baaf5f793d03d420 X-Google-Attributes: gid109fba,public From: ikastan@alumnae.caltech.edu (Ilias Kastanas) Subject: Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?] Date: 1996/09/28 Message-ID: <52j0j3$a3r@gap.cco.caltech.edu>#1/1 X-Deja-AN: 185817716 expires: October 4, 1996 references: <01 <52eha1$o7h@krusty.irvine.com> organization: Caltech Alumni Association newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada,sci.math Date: 1996-09-28T00:00:00+00:00 List-Id: In article , Jon S Anthony 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