From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Axiom of Choice
Date: 5 Jul 2000 10:50:18 -0500
Newsgroups: sci.math
Summary: [missing]

In article <8jus6e$qul$1@nnrp1.deja.com>,
Jim Heckman  <jheckman@my-deja.com> wrote:
>In article <8jujca$d07$2@cantuc.canterbury.ac.nz>,
>mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:


			..................

>> I hadn't been aware of this before, I'd thought that as soon as it
>> was identified (Zermelo 1904?) it became immediately accepted. Can
>> anyone enlighten us further on this matter?

>At least one of my textbooks claims that, in fact, when Zermelo proved
>the Well-Ordering Principle, mathematicians were so astonished that
>they looked for holes in his proof, and found: AC.

This was not the first explicit statement of AC.  Russell is
certainly earlier.

>BTW, in ZF+DC, what can be said about the cardinality of the real
>line R? It must still be Pow(aleph_0) =? 2^{aleph_0}, mustn't it?

This does not require anything more than ZF.  Some of us,
including myself, have even posted an explicit 1-1
correspondence, using continued fractions.

	I'm
>sure I've read that it's possible to prove that R is uncountable using
>only its order properties,

A complete dense linearly ordered set with no first or last
element and a countable dense subset is isomorphic to the
reals.  Other conditions can be used.  

	as opposed to the way it's usually done
>using its algebraic properties, via infinite binary series, etc.

The proof is not that much different.

	And
>I think I've read it's possible to prove that Pow(c) (=? 2^c) > c for
>any cardinality c, even without AC.

This is essentially the Russell paradox.

	Does the existence of uncountable
>well-ordered sets, which I *know* can be shown without full AC, depend
>on DC?

It depends on nothing beyond ZF.  The Hartogs function,
the set of all equivalence classes of well-ordered sets
no larger than a given set, is not as small as that set.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558