[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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