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: Tue, 22 Dec 1992 19:54:44 GMT

In article <1992Dec22.013357.18774@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:

[...]

>
>
>RH:
>>[...]
>
>What, Randall, an actual ellipsis?

I do it all the time.  But your words are usually worthy of extensive
commentary.

[...]

>RH:
>>It is possible to work in ZFC without referring to classes at all.
>>There is no need ever to refer to super-classes, etc.
>
>I have no idea why you think that this observation would vitiate any of
>my points.  Sure you can work in ZFC without *referring* to classes at
>all, -- but only if you are a formalist concerned solely with the
>syntactical aspects of the theory, in particular ignoring all semantical
>considerations.  More precisely, if you truly espouse the Quinean
>extensionalism, you may eliminate all reference to metatheoretical
>entities like super-classes; however you will purchase this liberty at
>the cost of inscrutability of reference.

This is not correct.  The point is that there are no such objects as
classes in the universe of ZFC (in its role as part of the real world;
I am not a formalist).  It is not necessary to admit the existence of
such objects; all reference to such objects can be avoided in favor of
discussion of syntax.

>
>To continue the Quinian idiom, you are proud as Punch of the proxy
>function which injects the universe of ZFC into that of NFU(+AxInf+AC).
>However, given that your function is not a bijection, you have stopped
>short of effecting an ontological reduction.  Therefore I contend that
>your interpretation of ZFC in NFU is, in fact, a misinterpretation of a
>theory built on incommensurable premisses.  In view of the above
>considerations, you are in no position to refute my claim.

The proxy function embeds the universe of ZFC into the universe of a
strong extension of NFU; NFU + Inf + AC is not strong enough to
guarantee the existence of more than a model of bounded Zermelo set
theory.  The premisses of extensions of NFU and ZFC are not nearly as
incommensurable as you think; they are very similar theories, when one
gets down to it.  But (speaking as a partisan of NFU) the universe
discussed by ZFC is not the _whole_ universe (although it is a very
important part of the universe); there is some more stuff about.

[...]

>
>RH:
>>"For all x, x = x" is a well-formed and true statement of ZFC.  It
>>says something about every object without exception.
>
>Indeed.  Why do you think that this contradicts any of my claims?

Eh?  You claim was precisely that we could not quantify over
everything.

>
>On second thought, it does, in the following way.  Consider a set
>theorist who, perhaps unlike yourself, believes that ZFC has
semantics.

You know perfectly well that I think ZFC has semantics (in the sense
that terms of ZFC refer to real things and sentences of ZFC make
statements about the world).  I do not think that ZFC has semantics in
the sense that its usual interpretation is the theory of a "model" of
ZFC in the technical sense, and neither should you.

>Thus his version of set theory properly consists of a unified language,
>consisting of (say) the first-order L_{ZFC} and the first-order language
>of the model theory of the same, L_{M(ZFC)}. Now, the latter happens to
>contain a recursive truth definition for the former, as well as a a name
>for each of its terms and sentences.  (I emphasize that it is a certain
>kind of union of both languages, that our hero regards as comprising his
>Set Theory.  Were he an intensionalist, he would modify my list by
>adding a recursive *sense* definition for each term, which he would then
>use to express a truth definition.)  Clearly, the (Tarskian, to be sure)
>lesson of this situation is that our hero is not at liberty to regard
>the variables of L_{ZFC} as ranging over *everything*, including the
>terms and sentences of L_{M(ZFC)}, at least as long as he remains
>unwilling to render inconsistent his Grand Unified Set Theory.  At this
>point, he has two reasonable solutions to his predicament, one of which
>is a special case of the other.  For more details, I refer you to
>Church's 1975 JSL article I have cited elsewhere.

I'm afraid not.  You are confused.  Of course, you are conveniently
forgetting that he also needs the first-order language of the model
theory of his model theory, and so forth, on your account.  But my
account is different.  No language other than L_{ZFC} is needed.  The
semantics of ZFC can be _expressed_ quite adequately in ZFC itself;
the facts about the semantics of ZFC are theorems which cannot be
proven from the axioms of ZFC (the additional axiom Con(ZFC) would be
needed, or, more conveniently, the existence of an inaccessible
cardinal) but that's fine; neither of us thinks that the full truth
about set theory is contained in any recursively enumerable set of
axioms).  Again, one also needs Con(ZFC + inaccessible) or a related
large cardinal hypothesis, and so on.  No paradox follows from the
model theory of ZFC being fully describable in ZFC and the quantifiers
of ZFC ranging over all its objects; consider that L_{ZFC} expresses
the notion "model of ZFC" without the least difficulty, and that (if
Con(ZFC) is true, as, of course, it is in the real world) the whole
theory of models of ZFC is subsumed in ZFC itself (as an NFU partisan,
I would observe that there are some big models of ZFC which it itself
cannot refer to, but that's not a serious problem for you).

[...]

>MZ:
>>>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...
>
>RH:
>>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.
>
>Please elaborate.  "Nothing can be said truly of what does not
exist."

One can talk about predicates as parts of sentences (syntactical
objects) without postulating objects to which they refer.  Some but
not all predicates (in the syntactical sense) have corresponding sets,
to which they can be said to "refer" by an abuse of terminology.  The
grammatical role of a predicate is not to refer to anything.

[...]


>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.
>
>MZ:
>>>Vide supra.
>
>RH:
>>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.
>
>Randall, I do not recall having contested unrestricted quantification in
>*your* sense

What do you think my sense is?

; I am merely arguing that taking seriously the *semantics*
>of ZFC, whose existence you are evidently prepared to deny, makes such
>quantification depend on a Tarskian hierarchy of classes, super-classes,
>etc....  Recall that any quantification over the objects of a Tarskian
>theory, must necessarily exclude from its range the objects particular
>to its metatheory.  This claim is so simple and so uncontroversial, that
>I wonder why you would bother to challenge it at all.
>
>Please explain the terrible logical flaws contained in my argument, or
>concede my point.

You are wrong about ZFC (or theories in general) not being able to
express (as opposed to prove theorems about) their model theory.  Even
second-order arithmetic can express its own model theory.  No
paradoxes ensue.  You are wrong in supposing that quantifying over a
domain of objects involves any reference to a set over which the
quantification takes place; it is odd that you, the ZF partisan, are
making this mistake and being corrected by me, the NF partisan (since
I do have such a domain ready to hand and you do not).

I think your plaint about semantics is that you are supposing that the
actual universe of ZFC is a model of ZFC.  But it is not; as you know
perfectly well, the Absolute Infinite is not a completed totality.
The actual universe of ZFC is not a model, because it is a fiction
(even for me -- my universe is a set, but its subclass which is the
universe of ZFC is proper).

[...]

>RH:
>>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.
>
>One man's illicit assumption is another man's analytic truth.
>Inscrutability of reference strikes again!

The Axiom of Replacement is NOT an analytic truth, not even for the
notion of set which you espouse.


Me:
>>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).
>
>Again, your sense of being "inside the theory" is quite alien to my
>theory of language.  This is a basic point of disagreement between us,
>and your failure to acknowledge it only supports my charges of a Quinian
>_advocatus diaboli_.

The sense of "inside the theory" here is simply "taking the theory
seriously on its own terms (as describing the whole world)".  It is
odd to take ZFC seriously in this sense, because it maintains that
everything without exception is a set, but ZFA can be taken as a
complete theory of everything (it is not the one to which I actually
subscribe, of course).

[...]

>
>Show me.  Describe a model of NFU, which is *philosophically*
>competitive with the iterative hierarchy, a.k.a. the intended
>interpretation of ZFC.  Keep in mind that I have successfully explained
>the latter to audiences whose average mathematical background did not
>exceed the curriculum of a semi-decent American (!) high school.  Should
>you succeed in doing so, I promise to renounce all claims of superiority
>of the limitation of size doctrine.

Your appeal to explicability to naive audiences is not valid, as you
know perfectly well; there is no reason why the real nature of the
world should be easy to explain to naive audiences.  The limitation of
size doctrine might be superior in terms of explicability, and still
be of limited applicability.

Asking for a _model_ is disingenuous (although you may not realize
this yourself, given your peculiar views on semantics); the intended
interpretation of ZFC is not a model of ZFC, since its domain is not a
set and its predicates are not relations (in the set-theoretical
sense).  The intended interpretation of NFU does slightly better since
its domain (the universe) and one of its predicates (equality) are
sets, but it still isn't a model, since the predicate of membership is
not a relation (not a set).  Again, I feel that it is strange that I
am instructing you about this, O extoller of the wisdom of Cantor.

The philosophical explanation of stratified comprehension is something
I've already addressed to you.  Essentially, the language of set
theory allows us to express predicates which are not genuine
properties of extensions; an extension is a collection of objects
which is in turn "identified" with an object, and the reason why "x is
not an element of x" (for example) is an illegitimate property of the
set x in that it exploits the relation between the object which _is_
the extension and the elements of the extension, which is not a
property of the extension _per se_.  It isn't in competition with the
iterative hierarchy; it subsumes it in its account of isomorphism
types of well-founded extensional relations.  The construction of
models of NFU in the usual set theory exploits the iterative
hierarchy; again, I've already described it to you.  The philosophical
merits of the Fregean definitions of cardinal and ordinal numbers,
recovered in this approach, should be quite clear.

A model of NFU is obtained by considering a model of an initial
segment of the iterative hierarchy which has an external automorphism
which moves an ordinal a to a larger ordinal j(a).  The domain of the
model of NFU is V(a) (the ath stage of the iterative hierarchy); the
membership relation "x E' y" of the model of NFU is defined as "x E
j(y) and j(y) E V(a+1)".  Verification that this is a model of NFU is
left as an exercise.

This is a rather complex construction; the reason for this is that it
is necessary to model a strict extension of the universe of the usual
set theory inside the usual set theory.  It is interesting to observe
that the interpretation of set theory of the usual kind inside NFU is
much easier.


>
>>-- 
>>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
>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Le cul des femmes est monotone comme l'esprit des hommes."
>


-- 
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
Xref: sparky sci.philosophy.tech:4627 sci.logic:2484