[FOM] The liar and the semantics of set theory

[Roger Bishop Jones]

On Friday 20 September 2002  1:21 am, Rupert McCallum wrote:
> --- Roger Bishop Jones <rbj at rbjones.com> wrote:
> > The liar "paradox" was used by Tarski to show that
> > arithmetic truth is not definable in arithmetic.
> >
> > It is natural to suppose that a similar result obtains in
> > relation to set theory, viz. that set theoretic truth is not
> > definable in set theory.
> > However, I am not aware of any argument (using the
> > liar or otherwise) which conclusively demonjstrates
> > this conjecture.
>
> Tarski's argument may be generalized as follows.
>
> Theorem. (Undefinability of truth).

It looks like you didn't read the rest of my message.
If I understand you correctly, you are showing at length
what I conceded, viz that if the semantics is given by
a single interpretation then a liar=like construction will
show that truth is not definable.

However, my question is about what happens if we
give set theory a loose semantics by stipulating a set
of more than one "intended interpretation" (e.g.
all standard models).

Then any sentence which does not have the same
truth value in all these interpetations does not have
a truth value under this semantics and nor does its
negation, so the liar construction fails to show that
this method of defining set theoretic truth cannot
be expressed in set theory.

Roger Jones