From: hrubin@b.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: Uncountability Question
Date: 12 Dec 1998 16:09:24 -0500
In article <3670bc25.6939653@news.csuohio.edu>,
Brian M. Scott wrote:
>On Thu, 10 Dec 1998 22:19:08 +0000, Jeremy Boden
> wrote:
>>In article <366f4922.4429351@news.csuohio.edu>, Brian M. Scott
>> writes
>>>In the treatment that is now customary, aleph_0 *is* a set, namely,
>>>the set w.
>>I'm confused now!
>>Why is a cardinal a set?
>As Herman Rubin points out elsewhere, it's not necessary to define
>cardinal numbers to be particular sets, and in the absence of the
>axiom of choice it probably isn't possible.
There is a way which works in ZF, and in many models of ZFU. There
are propositions equivalent to AC in ZF, but not in ZFU.
So if there are no individuals, define the rank of a set to be the
smallest ordinal number larger than the ranks of all its elements.
If there are, assume that there is a way of assigning rank to
individuals such that the class of individuals of a given rank is
a set (in NBG) or exists (in ZF). Then one can take the cardinal
number of a set to be the set of sets of the smallest rank which
can be put in one-to-one correspondence with it. This "definition"
is due to Dana Scott; it is a general way of selecting a set out
of a proper class in NBG, which is a conservative extension of ZF.
--
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