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: 109d8a,ae69c50ef02cd1c0,start X-Google-Attributes: gid109d8a,public X-Google-Thread: 109fba,baaf5f793d03d420 X-Google-Attributes: gid109fba,public X-Google-Thread: fc89c,97188312486d4578 X-Google-Attributes: gidfc89c,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public From: adam@irvine.com (Adam Beneschan) Subject: Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?] Date: 1996/09/26 Message-ID: <52eha1$o7h@krusty.irvine.com>#1/1 X-Deja-AN: 185537493 references: <01 <1996Sep24.133312 organization: /z/news/newsctl/organization newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada,sci.math Date: 1996-09-26T00:00:00+00:00 List-Id: jsa@alexandria (Jon S Anthony) writes: > >Actually, when people use the side-ways "8" notation, the typical >intent is simply that of indicating "arbitrarily large" or (somewhat >less so) denumerably infinite (size of naturals - aka Aleph0). Your >"Aleph2" is a denotation for the size of sets the size of the power >set of R (the reals) where Aleph1 denotes the size of sets >equinumerous with R. In general you have this whole "backbone" of >transfinite numbers constructed (typically) via the power set >operation. As long as we're getting pedantic: Is it really true that Aleph1 denotes the sets equinumerous with R? What I learned was that Aleph0 denoted the cardinality of N (the natural numbers), but the designation for the cardinality of R was the letter C (and I think the designation for the cardinality of the power set of R was the letter F). It's natural to ask whether C and Aleph1 are the same (where Aleph1 is the "successor" cardinality to Aleph0--roughly speaking, the cardinality of the smallest set that is larger than N); but from what I've read, it's been shown that the truth of the statement C = Aleph1 is independent of the other axioms of set theory. I asked sci.math about this several years ago, and I don't really remember what the answers were (I may have printed them out and stuffed them in my files at home), but I think they were along the lines of: The axioms of set theory don't really give you a definition of what a 'set' is, but if you come up with a suitable definition that satisfies all the axioms, you may be able to determine from your definition whether C=Aleph1 is a true statement or not. However, no one told me whether it was known whether C=Aleph1 or not in the "usual case" of the real numbers and integers. I believe the question is equivalent to the following: Is there an infinite subset of R that is not equivalent (equinumerous) to either R or N? My impression is that the answer may depend on how you define the concepts "real number" and "natural number" and how you define the sets R and N. I don't know whether there's a definite answer . . . I'm crossposting this to sci.math; maybe one of them can enlighten us. (Sorry to all the comp.* readers for the off-topic discussion, but now I'm curious.) -- Adam