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

[Rupert McCallum]

--- Roger Bishop Jones <rbj at rbjones.com> wrote:
[...]

> Someone with those beliefs might wish to define truth
> in set theory either as truth in all models of ZFC (and
> falsity as falsehood in all models of ZFC, leaving some
> sentences without truth value) or equivalently as
> provability in ZFC (i.e. a sentence is true if provable in ZFC,
> false if refutable in ZFC and otherwise
> has no truth value).

This semantics has the desirable properties

(1) phi is *true iff ~phi is *false
(2) phi and ~phi are not both *true

[...]


> I don't myself regard this account of the semantics of
> set theory as acceptable, and I don't believe that an
> acceptable semantics will turn out to definable in its
> own language.
> However, its not at all clear to me how to rule out the
> possibility that there is an acceptable semantics for set
> theory which is definable in the same set theory.
> 
> The challenge is then to come up with some (partial)
> criterion of acceptability for a semantics of set theory
> together with a proof that a semantics satisfying the
> acceptability criteria is not definable in a set theory
> having that semantics.
> 
> Bivalence would probably be a sufficient criterion.

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). We
could then define phi to be *false iff phi is false in V_kappa for at
least one inaccessible kappa, then we would have property (1) but not
(2). Or we could define phi to be *false iff it's not *true, then we
have neither property (1) nor (2). In this latter case, despite
bivalence we have the following:

Say that x *satisfies phi iff for some inaccessible kappa with x in
V_kappa (since in fact the x's we consider will be Goedel codes, this
proviso will be redundant) x satisfies phi in V_kappa.
Say that phi is a *truth predicate iff we have that x satisfies phi iff
x is the Goedel code of a *true sentence.
Say that phi is a *truth *predicate iff we have that x *satisfies phi
iff x is the Goedel code a *true sentence.
Define "*falsehood predicate" and "*falsehood *predicate" in the
obvious way (remember that "*false" means "not *true").

Then, letting phi(x)="x is true in V_kappa for some inaccessible kappa
with x in V_kappa", we have:

(1) phi is a *truth predicate and ~phi is a *falsehood predicate
(2) assuming there is no largest inaccessible, phi is a *truth
*predicate (the liar paradox is blocked by the failure of property (1))
(3) by the liar paradox there is no *falsehood *predicate.

I suspect you would want to have properties (1) and (2) as further
constraints to put on your alternative semantics.

[...]


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