Re: How to do magic with infinity

From: Han de Bruijn (Han.deBruijn_at_DTO.TUDelft.NL)
Date: 10/29/04 

Date: Fri, 29 Oct 2004 13:18:02 +0200

*** T. Winter wrote:
> In article <cliitn$9iq$1@news.tudelft.nl> Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> writes:
> > I don't know what exactly you have in mind. But an example of two sets
> > with intersection 0 is the set of all odd naturals and the set of all
> > even naturals. Both are taken from the set of all naturals. The latter
> > is FINITE to begin with. And is then allowed to grow to infinity. Thus
> > we must always start with a finite set, let it be {1,2,3, ... ,N}.
> >
> > If N is even then #odd = N/2 and #even = N/2
> > If N is odd then #odd = (N+1)/2 and #even = (N-1)/2
>
> Now let us build the set of natural numbers in a different fashion.
> At step N we add to the set we have:
> if N = 0 mod 3: 2N/3
> if N != 0 mod 3: 2N-1-2.floor(N/3).
> So we build the set of natural numbers in the sequence:
> {1, 3, 2, 5, 7, 4, 9, 11, 6, 13, 15, 8, ...}
>
> > We cannot really talk about the cardinalities of these sets as N -> oo .
> > All we can talk about is the ratio of these cardinalities as our set of
> > naturals approaches infinite size:
> >
> > #odd N/2 or (N+1)/2 1
> > lim ----- = lim -------------- = - = 1
> > N->oo #even N->oo N/2 of (N-1)/2 1
> >
> > Therefore "all" even and "all" odd naturals "have equal cardinality",
> > so to speak.
>
> And with my construction there are twice as many odd naturals as there
> are even natural (the limit of the quotient of the cardinalities is 2).
> Therefore the do not have equal cardinality so to speak?

I used the term "so to speak" on purpose. Because I'm well aware
of the fact that using separate "cardinalities" is slippery in
this context: only a quotient of "cardinalities" is meaningful.

An analogous situation occurs in classical calculus, where the
infinitesimals dx and dy have only a definite meaning as dy/dx.
They can only be used separately, as dx and dy, with caution.

But, for the rest ... YOU ARE RIGHT !

Your argument is both ingenious and valid. Of course, you are free
to use differently ordered sets of natural numbers instead. (To be
honest, I didn't think of that possibility.)

My argument has been sloppy. I should have realized that sets are
collections without an intrinsic ordering. I should have employed
the words "ordered sequence" instead of just "set".

Now I hope there is no misunderstanding about the meaning of "the
ordered sequence of natural numbers from 1 to N", as my "set".

Thank you for pointing this out.

Han de Bruijn

Quantcast