Re: Relation between sets and their elements

> Chris Menzel wrote:
> > Paul Holbach <paulholbachSPAMBAN@xxxxxxxxxx> wrote:

> > Given that both the Fs and the set of Fs exist,
> > are the Fs and the set
> > of Fs one and the same thing or not?

> The result of replacing occurrences of "The Fs" in a sentence with
> occurrences of "The set of Fs" yields a sentence with a different
> meaning and often a different truth value. "The planets
> revolve around
> the sun" is true, "The set of planets revolves around the sun" isn't.
> "The movers dropped the piano" is (let's suppose)
> true; "The set of
> movers dropped the piano" is false; sets can
> neither lift nor drop a
> piano. This strongly suggests that the Fs and
> the set of Fs are not the same.

Sounds plausible.

Graham Priest writes:

"If [the totality of cookies] is thought of as something independent of
the cookies, this may again sound implausible. If you are talking about
certain things, and if all existences are distinct, then invoking the
existence of another entity would seem de trop. This is the rhetorical
strategy employed in the last sentence quoted, 'It is one thing for
there to b e certain objects; it is another for there to be a s e t, or
set-like object, of which those objects are the members.' [Cartwright]
But the set and its members are not distinct existences. There could be
no set of cookies if there were no cookies--and vice versa. These are
no atomic, independent existences."

[Priest, Graham (2002). /Beyond the limits of thought/ (2nd ed.).
Oxford: Oxford University Press. (p. 282)]

I don't quite understand whether Priest means that the cookies and the
set of cookies are one and the same thing ("are not distinct
existences") or that they are two mutually (?) dependent things ("These
are no [...] independent existences"). (Of course, two identical things
are not independent of each other.)

If, as you argue, the Fs and the set of Fs are two different things,
then do the Fs depend on the set of Fs or does the set of Fs depend on
the Fs, or is there a state of mutual dependence?

Priest appears to hold the latter:
"There could be no set of cookies if there were no cookies--and vice
versa."

I think he's right insofar as the individual existence of the set of Fs
(as distinct from the empty set) depends on the existence of Fs, of
the Fs.

Formally:

["ixxFxx" = "the Fs"]

E!{x | Fx} -> E!ixxFxx

But is

E!ixxFxx -> E!{x | Fx}

true too?

If the sets exist, does the set of sets exist too?
Most probably not!

By the way, the plural reference of "the sets" seems indefinite, if the
sets don't exist as a complete multitude.

Priest obviously claims that

E!{x | Fx} <-> E!ixxFxx

is true.

But one should add that Priest, as a dialetheist, is prepared to
consider Cantor's inconsistent multitudes existent sets.

Regards
PH

.