Re: Resolving the paradoxes of set theory

From: George Greene (greeneg_at_cs.unc.edu)
Date: 11/02/04 

Date: 2 Nov 2004 15:18:27 -0800

ntspam2@netscape.net (Nathan) wrote in message news:<1e068d95.0411010859.79b3e535@posting.google.com>...
> daryl@atc-nycorp.com (Daryl McCullough) wrote in message news:<cm32ud0pce@drn.newsguy.com>...

> This is very nice, but without the power set axiom, what does it
> really tell you? Cantor's Theorem is the justification of the
> existence of unequal infinite cardinalities. If there is no set
> of all subsets of x, then what application can this theorem have?

That depends on what CLASSES also exist.
If you take (as JP and Darryl do) the position
that "the class of all x such that phi(x)" is
legitimately "writable" for EVERY phi (it is spelled
{x|phi(x)}, then this theorem (restated correctly, as
I did in another message) simply states that no
functional-class is onto the powerCLASS of its domain-class.
But not every class actually HAS a powerclass.
You cannot have a class of all classes that don't contain
themselves -- you have to settle for a class of all
SETS that don't contain themselves. This means you have
to say something in advance about what is a set and
what isn't. It is true
that for every class y, you can talk about
{x | x is a subclass of y}, but if y is not a set,
then x may not be either. The question arises as to
how big a universe you are quantifying over in the
"{x|" part of the abstraction notation.

The moral of the story is that JP needs to quit pretending
like "class-talk is just a notational convenience" and
move on up to a first-order axiomatization of an actual
class theory, like NBG.