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

[Roger Bishop Jones]

Let me begin this message with a recapitulation.

The thread begins with my doubts about the solidity
of the usually accepted view that the semantics of set
theory is not definable in set theory.

My doubts arise from the consideration that the obvious
demonstration depends upon bivalence and therefore
will not be applicable if a reasonable but non bivalent
semantics is put forward.
I put forward "truth as provability in ZFC" as a non-bivalent
definition yielding a version of set theory able to define its
own truth conditions (my conjecture, no so far disputed).

In addition to the liar the interpretation of modal logics
in set theory (as in "The logic o Provability" Ch.13) has been
discussed.
Such interpretations, if they were relevant, would presumably
argue for set theoretic truth being definable in set theory.
However, on my reading there is no case here of the truth
predicate of a language being defined in that language.
Boolos covers two of the sample semantics which I mooted
(truth as provabiliy, and truth as truth in standard models),
and the discussion has raised doubts about whether these
are reasonably candidates for definitions of the semantics of set theory.
I didn't understand the doubts raised, in particular the complaints
about what was or was not provable in the relevant modal
logics seemed to me not to provide a basis for discriminating
between interpreting box as "necessary", "provable" or "true".

However, since the provability and truth in standard models
interpretations are in other ways not wholly acceptable
candidates for a semantics of set theory, (e.g. no large
cardinal theorems are true in either), I would now like
to offer a single definition of set theoretic truth to provide
a more definite question.

The following definition treats set theoretic truth as a limit
to which interpretations of set theory approach more nearly
as they get larger.

TRUTH
A sentence S of first order set theory is TRUE iff there exists
a strongly inaccessible cardinal alpha such that for every
inaccessible beta >= alpha S is true in V(beta).

[a sentence is FALSE iff its negation is TRUE]

DEFINABLE
A set S of natural numbers is DEFINABLE in set theory if there
is a formula of first order set theory with a single free variable
F(x) such that for all n F(N) is TRUE iff n is in S (N the numeral for n).

PROBLEM
Is TRUTH (in set theory) DEFINABLE (in set theory)?

So far as I can see diagonalisation proves only that some
sentences are neither TRUE nor FALSE.

I'm interested also in what can be said for and against
this conception of set theoretic truth.

Roger Jones