[FOM] Truth and set theory
rgheck at brown.edu
Wed Dec 5 14:29:14 EST 2007
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.
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