From pycyn@aol.com Fri Jul 07 02:08:53 2000 Return-Path: Received: (qmail 26196 invoked from network); 7 Jul 2000 09:08:53 -0000 Received: from unknown (10.1.10.142) by m1.onelist.org with QMQP; 7 Jul 2000 09:08:53 -0000 Received: from unknown (HELO imo-d08.mx.aol.com) (205.188.157.40) by mta1 with SMTP; 7 Jul 2000 09:08:52 -0000 Received: from Pycyn@aol.com by imo-d08.mx.aol.com (mail_out_v27.10.) id a.f7.9d1b33 (662) for ; Fri, 7 Jul 2000 05:08:49 -0400 (EDT) Message-ID: Date: Fri, 7 Jul 2000 05:08:49 EDT Subject: Re: [lojban] 2 maths questions To: lojban@egroups.com MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: AOL 3.0 16-bit for Windows sub 41 From: pycyn@aol.com X-Yahoo-Message-Num: 3456 It is a bit of a mistake to think of infinity as a number and then say that natural numbers and even natural numbers have the same number of members, but, since we do it all the time, let's try to sort out what it means. In one sense, there is a boundary cardinal, aleph null, which is he cardinal of both the set of natural numbers and the set of even natural numbers (and the integers and the primes and.....). That means, ultimately, that, for each of these sets, there is a one-one mapping between that set and the canonical set that "is" (some funny mathematical sense) the cardinal. The mapping are different for the different sets, but can be combined to give one-one mapping diirectly between the different sets (naturals<=> evens, say). To say that there are twice is many naturals as evens is to say that there is a different mapping, this time between discrete pairs of naturals and evens (e with {e, e+1}, say). (Doing this for cases like the primes is a lot harder, for the integers or the rationals a lot fancier but not harder). Now, the fact that two things are related in one way does not usually mean they cannot be related in another as well; it seems paradoxical only in the case of size. But it is not a real paradox -- it just says there is a relation such that.... and another relation such that ... (and, in fact, it is not too hard to show that there are three of seventy-seven -- not hard but tedious -- times as many naturals as evens -- or, come to that, evens as naturals). It is like saying that somthing is green and tall: there are two standards where by the one.... and by the other ....