[FOM] The liar and the semantics of set theory

[Roger Bishop Jones]

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.

The situaltion is made a little more complicated by
there being some degree of uncertainty about exactly
what is the semantics of set theory.
The liar can be used to show that for a
large class of possible definitions of truth for set
theory, that definition cannot be carried through in
the set theory which has that semantics.

If truth of a sentence of first order set theory is defined
as truth in any single interpretation of that language,
the definability of this version of set theoretic truth in
the set theory with this semantics would allow the
construction of a sentence denying its own truth.

However, we might want to interpret first order
set theory as talking about all the "standard" models
of ZFC (i.e. V(alpha) for inaccessible alpha).

if truth of a sentence of first order set theory
is defined as truth in more than one "intended"
interpretation (and falsity as falsity in all of them),
then some sentences may lack a truth value
(even though they all have a truth value in every
interpretation).
In that case both the liar sentence and its denial
might lack a truth value and the proof that in such
a set theory its own set theoretic truth is not
definable would not go through.

Does anyone know of a conclusive argument
showing that set theoretic truth is not
definable in set theory?

Roger Jones
(with apologies if this question has already been covered)