🚀 go-pugleaf

In article <70f94e16.0111220954.5bd8439@posting.google.com>,
John <john_correy@excite.ca> wrote:
>torkel@scheutz.sics.se (Torkel Franz?) wrote in message news:<ul0bsi3cmx9.fsf@scheutz.sics.se>...
>> john_correy@excite.ca (John) writes:


>> > 1. UNRESTRICTED AXIOM OF EXTENSIONALITY OF Z SET THEORY

>> >    AxAy[Az(z in x <-> z in y) ->  x = y]

>> >    (Two sets are equal if, and only if, they have the same members.)

>>   Right, and if you add a predicate "x is a set" to the language of
>> set theory, you can make "if x and y are sets" explicit.

>kramsay@aol.commangled (Keith Ramsay) wrote in message news:<20011117095232.15000.00000541@nso-mr.aol.com>

>> One standard way of introducing "atoms" into a set theory is
>> by allowing the existence of a bunch of things that have no
>> members, and are distinct from each other (only one of which
>> is a set: {}).


>If the predicate "x is a set" is added to the language of set theory
>and its domain enriched as per above, how does one guarantee that one,
>and only one, non-membered thing is a set? By adding an axiom to that
>effect?

The usual method is to start out with a set, or class, of
individuals.  These are declared to be individuals, and
they are declared to have no elements.  Any of the usual
axiomatizations, modified to allow them, will prove that
there is an empty set in addition which has no elements.





-- 
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