[FOM] Truth and set theory

[Richard Heck]

praatika at mappi.helsinki.fi wrote:
> Dear FOMers
>
> Let us denote ZF (NBG) without the axiom of infinity as ZF- (NBG-). Z is
> the original Zermelo set theory, without the axiom of replacement. 
>
> Now apparently Wang (1952) showed that it is possible to give a (materially
> adequate) truth-definition for the infinitary Z in finitistic set theory
> NBG-. This is interesting enough.
>
> But is there some reason why one could not also give a truth definition for
> ZF, again in NBG- ? (Z and ZF have, after all, the same language) 
>
> Is anyone here familiar with this stuff?
>   
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/. And 
certainly NBG can produce a truth-definition for the language of set 
theory and prove all the T-sentences. Indeed, NBG is equivalent to ZF 
plus a Tarski-style truth-definition, with T(x) excluded from the axiom 
schemata. It wouldn't be shocking if NBG- were adequate, since infinite 
sets don't seem to play any role in the truth-definition itself. Some 
sets are needed for the coding of sequences and such, but only enough to 
get us a pretty minimal amount of arithmetic.

The situation is the same as with arithmetic and ACA_0. And, in fact, I 
believe that ACA_0 is far stronger than is actually needed for the 
truth-definition. A fairly weak arithmetic augmented by predicative 
comprehension should be enough for a truth-definition adequate for the 
proof of the T-sentences. I'd have to look back, but I seem to remember 
an earlier discussion here, or in a thread that went offline from FOM, 
in which some of us managed to convince ourselves that even Q plus 
predicative comprehension would do. But if not Q, then not much more 
than Q. Most of what you need for the truth-definition is just basic 
syntax, and the ability to express what is expressed in the language 
under discussion. But the former doesn't need much, and the latter is 
trivial except for the quantifiers, which is what predicative 
comprehension lets you handle.

That said, there is of course also the notion of truth in a model, and 
producing a truth-definition adequate for a theory T might be identified 
with constructing a model for T and showing that all theorems of T are 
true in the model. I don't know Wang's paper, but it would at least make 
sense for NBG- to be able to construct a model for Z without its being 
able to pull off the same trick for ZF.

Richard