[FOM] Proli's Question about Set Theory

Richard Heck rgheck at brown.edu
Sun Feb 26 22:06:23 EST 2006

```A.P. Hazen wrote:
> Andrea Proli notes that:
>
>> ... ZF is a first-order theory, and first-order
>> theories have a standard denotational, model-theoretic semantics. In model
>> theory, symbols are given an interpretation in terms of sets and relations
>> (which are also sets). Isn't this a circular definition?
>>
> ---There is certainly something  funny, at  least  on the  usual
> (sloppy) formulations.  Model theory texts say the domain of a model
> is a set, and ZF implies that there  is no set of all sets.  ((And no
> set of ordered pairs that can serve as the interpretation  of the
> membership predicate.)) So it certainly appears  that  the  "intended
> interpretation" of ZF is NOT, in the technical sense, a model  of it!
>
Further to these issues....

George Boolos's well-known reflections on second-order logic were driven
by this kind of concern. This is perhaps clearer in the earlier paper,
"On Second-order Logic", than in the later paper, "To Be Is To Be the
Value of a Variable (Or Some Values of Some Variables)", but it is
reasonably clear in the later paper, too.

There has been some recent work on so-called "unrestricted
quantification", that is, on the logic of a universal quantifier that
is, by "definition", required to range over all the objects there are.
Obviously, there cannot be any "models" of such a language, in the usual
sense, since there is no set of all objects. Exactly what should be done
question, too. Some quick references, off the top of my head: Vann
McGee, "Everything"; Agustin Rayo and Tim Williamson, "A Completeness
Theorem for Unrestricted First-order Languages"; McGee's comment,
"Universal Universal Quantification"; Williamson, "Absolute Identity and
Absolute Generality"; Rayo and Gabriel Uzquiano, "Toward a Theory of
Second-order Consequence". (I know Harvey wrote something important on
this topic, too, but I can't find a reference, at least not quickly.
Harvey?)

It does not seem to have been considered, in this work, to what extent
the sorts of constructions characteristic of classical model theory can
be carried out in the sort of framework being proposed for
"unrestricted" languages. I have other worries about these frameworks
myself, but that's perhaps the one most related to the topics of this list.

Richard Heck

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