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

[Roger Bishop Jones]

On Monday 23 September 2002  1:47 am, Rupert McCallum wrote:
> In "The Logic of Provability", George Boolos shows how to investigate
> issues like those under discussion in this thread via techniques of
> modal logic. We can re-interpret the diamond of modal propositional
> logic to mean "true in some V_kappa with kappa inaccessible" and the
> box to mean "true in all V_kappa with kappa inaccessible". As Boolos
> discusses in Chap. 13 of the cited work, Solovay proved in 1975,
> assuming infinitely many inaccessibles, that the following is a
> complete axiomatization of the modal logic of these notions:
>
> (1) all tautologies
> (2) box(a implies b) implies (box a implies box b)
> (3) box(box a implies a) implies box a
> (4) box(box a implies b) or box((box b and b) implies a)
>
> Rules of inference: modus ponens and necessitation

Thankyou for this reference.
This is certainly of great interest and I will spend some
time on it.

On the briefest perusal however, its not obvious how this
helps to settle the question at hand.

Do you know whether Boolos, for any of the interpretations
of the modal operators which might be thought ot as
rendering a semantics for set theory, settles (positively
or negatively) the question whether those operators
are definable in the set theory interpreted as having
box as the truth predicate.

That completeness is provable tells us that these
modal logics are much less expressive than the set
theories which they are about.
We know that there can be no complete formalisation
of truth in set theory if non-standard interpretations are excluded.
(this is like the incompleteness of second order logic).

Roger jones