From iad@MATH.BAS.BG Thu Jul 06 23:51:50 2000
Return-Path:
Received: (qmail 11416 invoked from network); 7 Jul 2000 06:51:51 -0000
Received: from unknown (10.1.10.26) by m1.onelist.org with QMQP; 7 Jul 2000 06:51:51 -0000
Received: from unknown (HELO argo.bas.bg) (195.96.224.7) by mta1 with SMTP; 7 Jul 2000 06:51:49 -0000
Received: from banmatpc.math.bas.bg (root@banmatpc.math.bas.bg [195.96.243.2]) by argo.bas.bg (8.11.0.Beta1/8.9.3/Debian 8.9.3-6) with ESMTP id e676paS22464 for ; Fri, 7 Jul 2000 09:51:38 +0300
Received: from iad.math.bas.bg (iad.math.bas.bg [195.96.243.88]) by banmatpc.math.bas.bg (8.9.3/8.9.3) with SMTP id JAA12151 for ; Fri, 7 Jul 2000 09:51:31 +0300
Message-ID: <39657E2D.45E9@math.bas.bg>
Date: Fri, 07 Jul 2000 09:52:30 +0300
Reply-To: iad@math.bas.bg
Organization: Institute for Mathematics and Computer Science
X-Mailer: Mozilla 3.01Gold (Win95; I; 16bit)
MIME-Version: 1.0
To: The Lojban List
Subject: Re: [lojban] 2 maths questions
References:
Content-Type: text/plain; charset=us-ascii
Content-Transfer-Encoding: 7bit
From: Ivan A Derzhanski
John Cowan wrote:
> On Fri, 7 Jul 2000, Thorild Selen wrote:
> > What you really want to say is probably that the set of even
> > numbers is a _proper subset_ of the set of integers, so there
> > is certainly a well known name for this relation.
>
> Yes, but it isn't quantifiable. I want to able to say that
> the set of integers is twice as "thick" ("dense" is already
> used for a different property) as the set of evens [...].
And in the same way the set of all integers that aren't divisible
by 3 is twice as thick as the set of integers that are, although
neither is a subset of the other (their intersection is empty).
Given that one can't obtain a (de)finite number by dividing two
infinities, the best we can do is talk about
(*) |X \cap S| / |Y \cap S|,
where X and Y are our two sets and S is an unbroken subset
of the integers, S = {k | m <= k <= n} for some m <= n.
Do we then take the limit of (*) for n-m -> \infty?
> What I don't know is whether this notion of "thickness" can be
> extrapolated beyond the sets which are multiples of some integer.
It can apply to some such sets (the integers whose decimal representation
ends in either 1 or 9 are twice as thick as those ending in 0), but very
many interesting sets don't have a constant thickness.
--Ivan