From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: A note on Modal Logic that has nothing to do with Ikky Sex
Organization: Boise State University
Date: Mon, 21 Dec 1992 18:03:11 GMT

In article <1992Dec18.205843.18689@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>Randall, evidently smarting from having had to concede a point to a
>lowly perpetual undergraduate, valiantly jumps into the fray:

	I conceded a very small fraction of a point, actually...

>
>In article <1992Dec18.163009.28035@guinness.idbsu.edu>
>holmes@opal.idbsu.edu (Randall Holmes) writes:
>
>>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:
>
[...]
>
>If you want precision, then be precise.  References to proper classes
>are *only* eliminable in the object language, because they are confined
>to the metalanguage.  Consider, for instance, the discussion in the
>beginning of Kunen's _Set Theory_, I.8.  I agree with Church that any
>(genetic) set theory gives rise to the set, class, super-class, etc....
>hierarchy in one way or another; I differ from him in not regarding it
>as an absolute shortcoming, relative to type theory.  (Moreover, I
>suspect that some corresponding stratification obtains within type-free
>theories; but my ignorance prevents me from pressing this point further
>

It is possible to work in ZFC without referring to classes at all.
There is no need ever to refer to super-classes, etc.

[...]

>
>RH:
>>That's what "for all x, P" means; and it is certainly true that for
>>all x, x = x, and I just said it.
>
>You know better than that, Randall.  As a good Quinean (or are you?),

Merely a fellow-traveller.

>you should be aware that reference is inscrutable between theories,
>whilst intratheoretical reference depends on all of your cognitive
>baggage.  What you say may be meaningful to you, but it is utterly
>meaningless to all right-thinking ZF patriots.  Methinks you ought to
>reconsider ontological relativity...

"For all x, x = x" is a well-formed and true statement of ZFC.  It
says something about every object without exception.

>
>MZ:
>>>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...
>
>RH:
>>Predication does not involve application, because the predicate is not
>>necessarily understood to be an object.
>
>As you discuss predicates in metatheory, your singular terms commit you
>to recognizing them as objects, in the same way the formulation of
>first-order logic commits you to a second-order metatheory.  Methinks
>you ought to meditate on what there is...

On the contrary, predicates are best understood via incomplete
sentences in meta-theory (not "as" incomplete sentences; the whole
point is that predicates do not have to be reified, which is
fortunate, since some cannot be reified).  If one quantifies over
predicates, one is then forced to admit that one is referring to
something.

[...]

>
>RH:
>>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.
>
>Vide supra.

I haven't seen any relevant comment.  ZFC does indulge in the
definition of sets via unrestricted quantification.  If one deprives
oneself of this indulgence, one is stuck in "bounded Zermelo set
theory", which is essentially the theory of types.

>
>RH:
>>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.
>
>But the principle of separation is true, as a consequence of the Axiom
>of Replacement!  No assumptions here.  Look, we have been through all
>this before; why not agree to disagree, as I suggested above?

The axiom of replacement is also an illicit assumption in this
context.  The universe is not an inconsistent totality; it is
incompatible with the axiom of separation or the axiom of replacement,
to be sure.

[...]
>
>MZ:
>>>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 domain that quantifiers range over in ZFC is the class {x|x = x},
and, assuming the axioms of ZFC, this is an illegitimate totality.
There is nothing to indicate that the contextually relevant domain in
ZFC is _not_ the sum total of everything, and this is even less clear
in ZFA (ZF with atoms) (the Great Pyramid of Gizeh then does not stand
accused of being a well-founded set).  Quantifiers in ZFC are not
limited in any way which is relevant inside the theory, and there is
little reason to believe that there is anything outside the domain of
ZFC (or at least ZFA).

>
>>>>Jamie
>
>MZ:
>>>ObBook: Michael Hallett, _Cantorian set theory and limitation of size_.
>>>Reasonably accessible to the dedicated amateur, willing to look things
>>>up.
>
>RH:
>>It's a good book.  But there are other approaches.
>
>Indeed there are.  Unfortunately, either they are false, or just not
>sufficiently developed to offer a convincing philosophical alternative;
>at this time, I doubt that anyone can tell whether the former or the
>latter is, in fact, true.

I can.

>
>>-- 
>>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
>
>ObBooks: Anything by Quine.  A class act, even if it is dead wrong most
>of the time.

Hah!
[...]
> 
>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
Newsgroups: rec.arts.books