From: Mike Oliver <oliver@math.ucla.edu>
Subject: Re: Cardinals
Date: Fri, 07 Jan 2000 14:19:54 -0800
Newsgroups: sci.math
Summary: [missing]

hack wrote:

> In the presence of Foundation (which I believe is considered part of ZF) it
> may be possible to "pick" the set of smallest rank with a given cardinality,

Not the set of smallest rank, exactly, but the collection of all sets of
that cardinality that are of the minimum rank possible.  That's a convoluted
statement, so to avoid confusion:  Let X be a set, and let alpha be the
smallest ordinal such that there exists a Y of rank alpha with |Y| = |X|.
Then you can take the cardinal of X to be the set of all Z of rank alpha
such that |Z| = |X|.

The "=" above is not meant to be asserting identity between two objects;
I write " |Y| = |X| " as shorthand for "there exists a bijection from
Y onto X".

> but I'm a bit wobbly on whether that's an adequate definition in the absence
> of AC.

AC is irrelevant here.

Of course this is mostly a technical issue, not a conceptual one.  It makes
perfect sense conceptually to abstract the notion of cardinality out of the
equivalence relation "there exists a bijection from X to Y", whether or
not you can identify the notion with a set.
==============================================================================

From: Mike Oliver <oliver@math.ucla.edu>
Subject: Re: Cardinals
Date: Sat, 08 Jan 2000 12:42:43 -0800
Newsgroups: sci.math

Keith Ramsay wrote:
> 
> In article <3876668A.17545BDC@math.ucla.edu>, Mike Oliver
> <oliver@math.ucla.edu> writes:
> |Not the set of smallest rank, exactly, but the collection of all sets of
> |that cardinality that are of the minimum rank possible.
> 
> Don't they call that "Scott's Trick"? Works for
> isomorphism classes generally.

Just so, after Dana Scott.  One of those things that seeems technical and
inessential in retrospect -- but if we didn't know how to do it it
would complicate our lives enormously.