From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: A note on Modal Logic that has nothing to do with Icky Sex
Organization: Boise State University
Date: Fri, 18 Dec 1992 16:30:09 GMT

In article <1992Dec18.015229.18660@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>In article <1gr2fmINNl8s@cat.cis.Brown.EDU> 
>PL436000@brownvm.brown.edu (Jamie) writes:
>
>>zeleny@husc10.harvard.edu (Michael Zeleny) said:
>
>MZ:
>>>Finally, logicians quantify over all sets all the time; that's what
>>>the notion of a proper class is for.

Not precisely.  One quantifies over all sets in ZFC proper, in which
there is no essential reference to proper classes (all such references
are eliminable).

[...]

>
>J:
>>I suppose we can quantify over all the sets. But can we quantify over
>>everything at once? Intuitively we certainly can. "When I say
>>'everything,' I mean 'EVERYTHING.'" Didn't I just quantify over
>>everything at once? EVERYTHING at once?

That's what "for all x, P" means; and it is certainly true that for
all x, x = x, and I just said it.

>
>I have another one, which you may contest: any instance of predication
>*seems* to presuppose a universal applicability.  But we all know what
>the semantical antinomies do to that one...

Predication does not involve application, because the predicate is not
necessarily understood to be an object.

[...]

>
>J:
>>Does Zeleny think it is possible to quantify over everything?
>
>I think not.  "Everything" is an inconsistent multiplicity, which does
>not admit of reference, much less quantification.  (The reason we can
>understand this sort of discourse, is because it has the semantics of a
>figurative circumlocution.)  There is no such thing as the universe, or
>else if there is one, it cannot be designated.

{x|x = x}.

  (Am I sounding like
>Gorgias yet?)  Naturally, Randall Holmes will disagree, but I am
>speaking as a _bona fide_ ZFC chauvinist.

No, you aren't.  ZFC is irrevocably committed to quantification over
absolutely everything; if quantifiers in ZFC have to be restricted to
sets, one obtains a theory with the precise strength of the theory of
types; for example, one cannot prove the existence of aleph-omega.

The proof of the consistency of NFU relative to ZFC shows that
"everything" is not an inconsistent totality.  Get it through your
head.  Nothing speculative about it; this is cold, hard, mathematical
fact.  "Everything" inherits the inconsistent nature of the Russell
class if one assumes the axiom of separation; it is not in itself
paradoxical.

[...]

>
>I believe that it is implicitly qualified to the contextually relevant
>universe of discourse.  Which is not to say that one cannot quantify
>over all sets, or all classes, or all cumquats, or all pubic hairs,
>or...

The "everything" which is an inconsistent totality in ZFC is the
implicit universe of discourse itself, not some larger totality.

>
>>Jamie
>
>ObBook: Michael Hallett, _Cantorian set theory and limitation of size_.
>Reasonably accessible to the dedicated amateur, willing to look things
>up.

It's a good book.  But there are other approaches.

>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Nothing can be said truly of what does not exist."
>


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu