From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.num-analysis
Subject: Re: Infinity question... please help solve a bet!
Date: June 19, 1997
In article <33A84AA4.6B51@mci.com.bite.me.spammers>,
Mike Secorsky wrote:
>An individual I work with has made a bet with another co-worker
>concerning infinity. His claim is that there are an infinite number of
>infinities, and he uses the following example...
This is not really fodder for s.m.n-a. I've directed followups to
sci.math, where it'll keep 'em busy for weeks.
Yes, "infinite" is a concept rather than a quantity; it means "not finite"
and may be described, for example, as the ability to pair off the
elements of a set with the elements of a proper subset. Your example, for
instance, shows that the set of counting numbers is infinite, since
it can clearly be put in one-to-one correspondence with a subset of itself
(the set of _even_ counting numbers).
But if you wish to say more than "not finite", you may. Given two
infinite sets, you may ask whether or not it is possible to pair off the
elements of one set with the elements of the other. So while the set of
counting numbers is infinite, as is the set of even counting numbers, and
the set of all even perfect squares, these sets are all "equally large"
(i.e. the elements can be paired off) even though each is clearly a proper
subset of the one listed before it.
Are all inifinite sets "equally large" in this sense? No - for example,
one may prove that the set of all real numbers is so large that you can
never pair off _all_ its elements with the set of counting numbers.
So everyone's a winner here: yes, there are "different infinities" but no,
the proffered example does not suffice.
By the way, the fun is just beginning here, hence a large field of
mathematics known as set theory. The axioms usually used to clarify
what makes a set enable us to prove something rather remarkable: it's
possible to set up a consistent "ranking" of all sets so that
we can call some sets "larger" than others (Here I put this in quotes
because I don't mean that either of them has to be a subset of the
other; and moreover, as we have already seen, a set is not necessarily
"larger" than a proper subset -- the two may be equally large). Even
better, there will always be the concept of "the next biggest size".
Then we can call the "smallest" infinite size "aleph-0" (this would be for
any set which can be paired off with the counting numbers), the next
size "aleph-1", and so on.
None of this is remarkable if you only deal with finite sets but with
infinite sets stranger things happen. For example:
* There is not always the concept of "the next smaller size"
(e.g., all sets which are "smaller" than the set of counting
numbers are in fact finite, and there's no last finite size!
To give another example, if you have sets A0, A1, A2, ... of
sizes aleph-0, aleph-1, etc., then their union A_w has a brand new size,
but there is no "next smaller size" just smaller than A_w.)
* The usual axioms for set theory are insufficient to decide whether
or not there are "in-between" sizes in general. The Continuum Hypothesis
isn't so much a hypothesis but a question: about this first set A_1 which
is so big that it can't be paired off with the counting numbers -- is it
so big that it can be paired off with the set of all real numbers? Or is
there a set of intermediate size? Turns out the question is independent
of the other, more believable axioms for set theory. That is, the
answer to this question depends on what you mean by "set" !
* There's an infinite number of different sizes of infinite sets. If
you're still with me, you should be asking, "OK, so we have this
twisty maze of little labels, all different: aleph-0, aleph-1, etc.
Sure, the set of these labels is not finite. But can it be paired off
with the counting numbers? Or are there so many _labels_ that we can't
even pair them off with, say, the set of all real numbers?" Well, your
worst fears will be realized: the collection of all labels is so big
isn't even really a set...
Some folks say you should never attempt to think about set theory
while drinking; the theory is too disorienting as it is. But others
say that's the _only_ state of mind in which the subject can be approached!
dave