Re: Uncountable sets in CZF?
From: Acid Pooh (poopdeville_at_gmail.com)
Date: 09/14/04
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Date: 14 Sep 2004 15:54:03 -0700
agamemnon_atheos@yahoo.com (Agamemnon) wrote in message news:<88ee1d13.0408180222.574a4671@posting.google.com>...
> 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?
I assume that you here mean "undenotable"--we can only point to
countably many real numbers. This is correct. But you can do quite a
bit of work with only countably many real numbers--namely, you can
approximate any real number to any degree of accuracy. (This is the
basis for real analysis -- you might also be interested in the theory
of computable reals)
>Remember, it is impossible to come up with a
> counter-example to the claim "all reals are defineable."
Who needs a counter-example when you can prove it directly?
>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?
The set-theoretic notion of a bijection is an extension of the
intuitive notion. Notice that they coincide exactly in the finite
case if we imagine a function as an arrow which points from an object
in A to an object in B, where A and B are sets whose sizes we wish to
compare.
'cid 'ooh
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