Cantor's Infinities and Universal Sets

Date: 07/17/2008 at 03:27:17
From: Martin
Subject: Cantor's Infinities

In Cantor's set theory, the idea of having a universal set or a set 
of everything cannot be true, due to the basic contradiction that 
arises from the nature of set theory.  Based on this, when looking at 
Cantor's Hierarchy of Infinities, does the hierarchy of infinities 
still hold?  Truly, if an absolute infinity existed then it would 
accommodate everything...contradicting the idea of no universal set. 

Date: 07/17/2008 at 08:32:31
From: Doctor Tom
Subject: Re: Cantor's Infinities

Hi Martin.

You're right--you cannot have a set that, say, contains one infinite
set of each size or you'll run into the problem you mention.

One way around it, for the purposes of discussion of the idea, is to
use the idea of a "class" that can be that large.  The restriction, of
course is that if such a class is too large to be a set (in other
words, it can't be built from the allowed operations of set theory), 
then it is called a "proper class", and cannot be contained in any set
or other class.

That way I can talk to other mathematicians about "the class 
containing all the ordinal numbers", but I can't include that in any
set or class.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/18/2008 at 09:23:44
From: Martin
Subject: Cantor's Infinities

But the idea of Cantor's infinities is based upon looking at 
bijections.  For instance, the set of integers has a lower 
cardinality than the set of real numbers because a bijection between 
the two sets does not exist.  So Cantor looked at bijections to 
determine the size of infinities.  If so, are we stuck with looking 
at Sets?  In addition, could it be that there does not even exist a 
hierarchy of Infinities as Cantor has hypothesized?

Many thanks.  I'm doing a maths essay on Cantor and infinity so I just
thought of this...it would be great if you can offer any advice.

Date: 07/18/2008 at 10:35:34
From: Doctor Tom
Subject: Re: Cantor's Infinities

The problem with naive set theory, where you're sloppy about what you
allow to be a "set", is (Bertrand) Russell's paradox:  Consider the
set of all sets that don't contain themselves as an element.  Is that
set a member of itself or not?  If it isn't, it is, and if it is, it
isn't.

To get around this what's done is to carefully build up a collection
of valid sets, starting from the empty set, so that nothing you make
can contain itself.  In such a system, you can't have a universal set
(that contains all sets) or it would have to contain itself and you're
back to Russell's paradox.

The most common set of axioms used to build up a formal (and as far as
we know, contradiction-free) theory of sets is called the "Zermelo-
Frankel" axioms for set theory.

Those axioms include the "axiom of infinity" that tell us basically
that the natural numbers can be put in a set, and the "power set
axiom" that tells us that if S is a set, then the power set of S is
also a set.  With Cantor's work, we can show that a hierarchy of
infinities exists: a sequence of sets, each infinite, and each larger
than the last.  We cannot put ALL of those into one set, however,
since we'd be back at Russell's paradox.  But we, as mathematicians,
need to talk about "all the sizes of infinity", so when we do want to
talk this way, we certainly don't want to say, "the set of all the
infinite cardinal numbers" or we'll face Russell again.  So when we
talk about these larger groups, we just call them "classes", with the
understanding that a class cannot be made that contains another 
class, and classes cannot be members of sets.

Actually, the word "class" is a little looser, and a class could be a
set, but if it's too large, we just call it a "proper class", and so
to be very precise, I should say that no PROPER class can be contained
in another class or set.

I'm not sure how much you know about formal set theory or the
hierarchy of infinities, but I've written a couple of papers aimed at
the level of bright high-school students that you might find
interesting as background, although they don't treat your topic
directly.  Take a look at these:

  Infinity
    http://www.geometer.org/mathcircles/Infinity.pdf 

and

  Nothing
    http://www.geometer.org/mathcircles/nothing.pdf 

Good luck with your project!

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/