[FOM] Proli's Question about Set Theory

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Sat Feb 25 23:33:42 EST 2006

```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!

The model theory textbooks are right to define "model" as they do:
model theory, as a branch  of MATHEMATICAL logic, is a branch of
mathematics, and is typically conducted, these  days,  in the more or
less explicit framework of axiomatic set theory.  (There are people
on this forum -- Harvey Friedman and Stephen Simpson come to mind--
who have elaborated on this.  Various statements of model theory turn
out to require for their proof, and even [[modulo some weak basic
assumptions]], sometimes, are equivalent to, set-existence  axioms.)
So we are just STUCK with a foundational axiomatic system which, on
its primary and intended  application, is not interpreted in a model.
(We all hope that it DOES have other, unintended, interpretations in
models: Gödel's completeness  theorem tells us [[modulo some weak
basic assumptions]] that  having a model   is equivalent to being
formally consistent for First-Order theories!)

Set theorists seem not to be bothered by this, but it is a
curious conceptual situation worth some examination in the philosophy
of logic.  John Etchemendy's book on the "Concept of Logical
Consequence" (I think that's the title but not quite sure) touches on
it; his discussion takes off from Kreisel's in "Informal Rigor and
Completeness Proofs" (in Lakatos, ed., "Problems in the Philosophy of
Mathematics"; Kreisel's paper is partially reprinted in Hintikka,
ed., "Philosophy of Mathematics").
(Précis of central Kreisel-Etchemendy point: if you DEFINE a
valid First-Order sentence  as one that holds in all models,
it does not IMMEDIATELY follow that all valid sentences in
the language of set theory are true, since the intended
interpretation of set theory is not amodel.  There is a way
around  this for First-Order logic-- logically provable
sentences ARE truths of set  theory-- but there  are  hard
questions for more powerful formal languages: e.g.languages
with generalized  quantifiers.)

And Richard Cartwright's (characteristically clear!) paper "Speaking
of Everything" ("Nous" vol.  28 (1994) pp. 1-20) gives a forthright
defense of the position that it is not required, in order for a
First-Order theory to be
legitimate/acceptable/comprehensible/understood/true, that its
variables be interpreted as ranging over a  set.

---

Allen  Hazen
Philosophy Department
University of Melbourne

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