Re: Uncountable sets in CZF?

From: Agamemnon (agamemnon_atheos_at_yahoo.com)
Date: 08/18/04 

Date: 18 Aug 2004 03:22:12 -0700

fishfry <BLOCKSPAMfishfry@your-mailbox.com> wrote in message news:<BLOCKSPAMfishfry-324166.22494716082004@netnews.comcast.net>...
> In article <88ee1d13.0408161814.cec506a@posting.google.com>,
> agamemnon_atheos@yahoo.com (Agamemnon) wrote:
>
> > [snip]
> > "All this means is that if you define size (cardinality) a certain
> > way, and use these particular axioms to capture the logic of sets,
> > then it follows that uncountable sets exist. Don't read too much into
> > it."
> >
>
> Yes, but these axioms and definitions have a certain intuitive appeal.
> You agree that some infinite sets can be put into one-one correspondence
> with the natural numbers; and other infinite sets, can't. You must still
> regard that as strange and wondrous, no? Because the definitions and
> axioms are in no way artificial or forced.

It is interesting, but I wouldn't use the word 'wondrous'. I agree
that, in the real world, two collections have the same size if their
elements can be put into a one-to-one correspondence. However I don't
agree that my intuitive notion of a one-to-one correspondence which I
use in the real world is the same as a set theorist's notion of a
bijection in set theory.

In the real world, I can show a one-to-one correspondence of apples
and oranges by placing each apple next to exactly one orange. Within
set theory, I cannot place sets next to each other, or draw an arrow
from one set to another - the only way to show the correspondence is
by using other sets within the theory.

Given finitly many symbols, one can make only countably many
sentences, which means one can only define countably many real
numbers. But from within set theory you can prove that there are
uncountably many reals. So do we conclude that some reals are
undefinable? Remember, it is impossible to come up with a
counter-example to the claim "all reals are defineable." Or do we
conclude that our notion of a bijection within set theory isn't the
same as our intuitive notion of a one-to-one correspondence in the
real world, and so we cannot meaningfully speak about the relative
sizes of infinite collections?

Oh, and as for the axioms of ZFC not being 'artificial', I'm not so
sure about that. In a natural language such as English, people can
talk about collections without restriction, and this leads to various
paradoxes. I think our intuitive notion of a set is contradictory -
just like Cantor's "naive" set theory. In order to axiomatize the
concept of sets in a non-contradictory way, restrictions must be
placed on how we can make sets. But ZFC is only one way of doing this
and so it is, in that sense, artificial. Sure, the axioms of ZFC are
pretty intuitive, but so are the axioms of naive set theory.

Quine's NF trys to be more like naive set theory, and many theorems of
ZFC are false in NF (Cantor's theorem is one example). CZF, on the
other hand, is even more restrictive than ZFC. I suspect many of the
"paradoxes" of ZFC do not occur in CZF, but I'm not sure about this.
(the Banach-Tarski paradox is one example of the sort of paradox I'm
talking about)

-Agamemnon

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