🚀 go-pugleaf

In article <3BF86980.CC10AFAF@math.ucla.edu>,
Mike Oliver  <oliver@math.ucla.edu> wrote:
>John wrote:

>> Unfortunately, the meaning of "we will restrict extensionality to
>> sets" is hardly clear.  It could mean equi-membered things have to be
>> equal IF they are sets; it could mean equi-membered things have to be
>> equal ONLY IF they are sets; and it could mean equi-membered things
>> have to be equal IF AND ONLY IF they are sets.

>This paragraph doesn't make any sense to me.  The only possible
>sense of "restrict extensionality to sets" is, if you know two
>things are sets and have the same members, you can infer they're
>equal.  No assertion at all is made about things that have the
>same members but are *not* sets.  Obviously they *can* be
>equal, even if they're not sets -- a thing is equal to itself,
>whether it's a set or not.

The more customary approach to ZFU(C) is to consider that
some objects are NOT sets, but are available to be used as
elements.  These have no elements, but are not the empty set.
This can also be done with NBG.  All of the early models of
consistency of the negation of choice used these individuals,
or as they were called in German, Urelemente.
-- 
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