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

Roger Bishop Jones rbj at rbjones.com
Sat Sep 21 12:41:04 EDT 2002

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).

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).

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?
It does not appear that the liar paradox can be used to prove
that there is none, since the liar sentence might be true in
some of the specified models and false in others,
so if this conjecture is true, how can it be proven?

Roger Jones

More information about the FOM mailing list