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From: Jesse F. Hughes on 3 Jul 2010 17:40
Transfer Principle <lwalke3(a)lausd.net> writes:

> Note that Hughes was the first to use the word "satisfy" in
> this manner:
>
> Hughes:
> Even in a not-too-distant from standard set theory like NF, it
> seems fairly evident to me that objects which could conceivably
> satisfy NF are somewhat different from objects that would satisfy,
> say, ZFC.
>
> So what does Hughes mean for an object to "satisfy" a theory? I
> interpreted it to mean that there exists a model of NF which
> proves the existence of the object, but not one of ZFC, but
> maybe Hughes had something else in mind.

It would be nice if you would specify *which* Hughes you meant, since
there are two different Hugheses that are posting in similar threads.

--
Jesse F. Hughes
"He was still there, shiny and blue green and full of sin."
-- Philip Marlowe stalks a bluebottle fly in
Raymond Chandler's /The Little Sister/
From: MoeBlee on 3 Jul 2010 17:58
On Jul 2, 3:58 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote:
> On Fri, 2 Jul 2010 14:40:34 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com>
> said:
>
> > On Jul 2, 12:12 am, Transfer Principle <lwal...(a)lausd.net> wrote:
>
> >> at what point beyond which a theory differs from ZFC such that we
> >> should no longer call the objects which satisfy them sets?
>
> > I don't know. But we can adopt certain definitions, such as:
>
> > x is a set <-> (x=0 or Eyz y in x in z))
>
> > That will work as long as the theory defines '0' appropriately.
>
> And surely extensionality is essential to our conception of set.

As far as I can tell, it is, Chris.

MoeBlee

From: Transfer Principle on 6 Jul 2010 16:38
On Jul 3, 2:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Transfer Principle <lwal...(a)lausd.net> writes:
> > So what does Hughes mean for an object to "satisfy" a theory? I
> > interpreted it to mean that there exists a model of NF which
> > proves the existence of the object, but not one of ZFC, but
> > maybe Hughes had something else in mind.
> It would be nice if you would specify *which* Hughes you meant, since
> there are two different Hugheses that are posting in similar threads.

Apparently, neither. It was actually herbzet who made that
comment above. I'm not sure how I got confused and thought
that either Hughes made the comment that herbzet made.

I apologize for the confusion.
From: Transfer Principle on 6 Jul 2010 16:43
On Jul 3, 1:49 am, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote:
> On Fri, 2 Jul 2010 20:46:04 -0700 (PDT), Transfer Principle
> > Menzel gives some criteria, namely that it should at least adhere to
> > Extensionality, and that sets ought to contain elements (except 0)
> > and be elements of other sets.
> Minimally. But when one starts constructing a more definite picture,
> notably, the cumulative conception of sets, a lot more would seem to be
> required, e.g., foundation.

Foundation? In that case, would ZFA still be a "set theory,"
as it refutes Foundation/Regularity?

I knew that Extensionality was a biggie, since some of WM's
old ideas (especially wrt "potentially infinite" objects)
were said to be non-sets.

Also, I've heard that those posters who wanted to use
mereology (zuhair, galathaea, and tommy1729) also needed to
come up with names other than "sets" for their objects (yet
tommy1729 still insists on calling his theory "tommy1729's
_set_ theory).

Thanks for the info!
From: Chris Menzel on 7 Jul 2010 08:59
On Tue, 6 Jul 2010 13:43:56 -0700 (PDT), Transfer Principle
<lwalke3(a)lausd.net> said:
>> > Menzel gives some criteria, namely that it should at least adhere
>> > to Extensionality, and that sets ought to contain elements (except
>> > 0) and be elements of other sets.
>> Minimally. But when one starts constructing a more definite picture,
>> notably, the cumulative conception of sets, a lot more would seem to
>> be required, e.g., foundation.
>
> Foundation?

Well, yes, IF one insists upon basing one's notion of set on the
cumulative conception.

> In that case, would ZFA still be a "set theory," as it refutes
> Foundation/Regularity?

It would not be for those who would consider the cumulative conception
to be the only legitimate conception of set. But that would be a silly,
doctrinaire way to think. The well-founded and non-well-founded
universes simply reflect two related but different conceptions of set.
Both lead to rich and interesting theories. To try to determine whether
one or another conception corresponds more closely to some ordinary
intuitive notion of set might be a mildly interesting semantic or
anthropological exercise, but it doesn't seem to me to be a
philosophically or mathematically significant one.


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