[FOM] The liar and the semantics of set theory (expansion)

[Rupert McCallum]

--- Roger Bishop Jones <rbj at rbjones.com> wrote:
> On Saturday 21 September 2002  5:53 am, Rupert McCallum wrote:
> 
> > (1) phi is *true iff ~phi is *false
> > (2) phi and ~phi are not both *true
> 
> > In an earlier message I believe you suggested defining phi as *true
> iff
> > phi is true in V_kappa for at least one inaccessible kappa (note
> that
> > if the universe is pi-1,1-indescribable then truth implies *truth).
> 
> No I didn't suggest that.
> (This idea was first mooted by you in an off-list message.)
> The closest I came was to consider interpreting set theory as
> about all standard models, but that would require an ALL where
> you have an ANY (which makes a big difference).
> 

I see. Yep, definitely a much better idea. My apologies.

> If you say "at least one" then you get an unsatisfactory
> definition of truth.
> e.g. both a sentence and its negation might be true
> (e.g. the sentence "there exists an inaccessible ordinal"), 
> and the conjunction of two true sentences is not necessarily true.
> 
> > I suspect you would want to have properties (1) and (2) as further
> > constraints to put on your alternative semantics.
> 
> Or something stronger.
> 
> For example, suppose we require that the semantics be specified
> as some set of standard models of ZFC, so that truth of a sentence
> is truth in all the specified models and falsity as falsity in all
> the
> specified models.
> That's quite a strong constraint which I think implies your (1) and
> (2).
> 

Yes, it does.

> Is it then true that there is no semantics (satisfying the
> constraint)
> such that set theory interpreted according to that semantics
> can define its own semantics?

If the class of models in question is definable, there will be a *truth
predicate. You are of course asking whether there is no *truth
*predicate.

If bivalence holds, the liar paradox goes through and there is no
*truth *predicate.

If not, let S be true in one of the "standard" models M and false in
another, N. Then S is independent of ZFC, and, assuming ZFC cannot
prove its own inconsistency, Con(ZFC+S) is independent of ZFC. If the
class of models includes a model of ZFC+~Con(ZFC+S), then in this model
S will be *false. Thus, if we assume

(1) if something is *true, it's *true that it's *true
(2) If S is neither *true nor *false, then the class of models includes
a model of ZFC+~Con(ZFC+S)

then "*truth in all 'standard' models" is a *truth *predicate. 

On the assumption that ZFC cannot prove its own inconsistency, the
semantics you proposed where *truth equals ZFC-provability and
*falsehood equals ZFC-refutability satisfies these two conditions. As
you say, the liar paradox is blocked by failure of bivalence, so that
the sentence "I am *false" is neither *true nor *false. Indeed this is
precisely the point of Gödel's incompleteness argument.

It would be interesting to see what happens if we restrict ourselves to
well-founded models.

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