[FOM] Truth and set theory

Richard Heck rgheck at brown.edu
Fri Dec 7 15:11:09 EST 2007

praatika at mappi.helsinki.fi wrote:
> Quoting Richard Heck <rgheck at brown.edu>:
>> I'm somewhat puzzled by this question, for reasons close to ones you 
>> mention here in passing. Namely: A theory (or definition) of truth is a 
>> theory (or definition) of truth for a /language/. I don't know what one 
>> would mean by saying that one had defined truth for a /theory/.
> Yes, this puzzles me too. (Though, Tarski allowed that a "language" may
> contain some non-logical axioms). I just wonder why Wang's result is always
> stated for Z, and not for ZFC. Just an historical accident?
> I would expect that if one can give in NGB- a materially adequate truth
> definition for  ZF-, this would automatically be a truth definition for Z,
> ZFC and whathever shares the same language. Right?  

One would have thought so.

That said, in the paper to which Kremer referred, by Ray, there is some 
discussion of this, and it looks as if at least some people use the 
expression "giving a truth definition for a theory T" to mean something 
like: prove the existence of a model for T. In Ray's discussion, for 
example, he considers a case where one defines "truth for Z", and what 
this actually involves is defining truth for Z with the quantifiers 
restricted to V_{\omega + \omega}, where of course you now need axioms 
strong enough to prove the existence of that set. One can then go on, of 
course, to show that the axioms of Z are true in said model, etc, etc.

This is of course something one can do, but to call it "defining truth 
for Z" seems to me to invite confusion. Indeed, Ray's defense of Tarski 
on essential richness would have been much stronger, IMHO, had he simply 
insisted that this isn't a "definition of truth" in the sense that is 
relevant there. It's worth noting, indeed, that Tarski's paper on truth, 
CTFL, was written before the paper on logical consequence, and, in 
famously, even in the latter Tarski doesn't consider domain variation: 
"all" just means all, and Z is a theory about the sets, not about some 
of them, but about all of them.

Frankly, I think Tarski was right. If you want a theory about V_{\omega 
+ \omega}, you can have that, but the right way to formalize it is by 
explicitly relativizing the axioms of Z by means of a predicate letter D 
whose intended interpretation is V_{\omega + \omega}. By no means does 
this imply that you can't consider other domains for the quantifiers 
when doing model theory. Model theory is one thing, because truth in a 
model is one thing; and truth simpliciter is a different thing. Nor does 
it imply that, in studying Z, you can't more or less ignore the niceties 
and work with the usual axiomatization. We have theorems to that effect, 


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