Subject: Re: [lojban] 2 maths questions

From: Ivan A Derzhanski <[email-masked]>
Date: Fri, 07 Jul 2000 09:52:30 +0300

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