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

Rupert McCallum rupertmccallum at yahoo.com
Tue Oct 1 18:32:06 EDT 2002

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

Yes, this conjecture is correct. If S is ZFC-provable, then it is
ZFC-provably ZFC-provable, and the converse holds provided ZFC is

So, briefly adopting my old usage, if you define "*true" to mean "true
in all models of ZFC (i.e. ZFC-provable)", and "x *satisfies phi" to
mean "x satisfies phi in some model of ZFC (i.e. ZFC proves phi(x))",
and say that phi is a *truth *predicate if we have that x *satisfies
phi iff x is the Goedel code of a *true sentence, then we have that "is
ZFC-provable" is a *truth *predicate, by the above.

> 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.

Yes, there is. Assuming infinitely many inaccessibles, "is *true" is
easily seen to be a *truth *predicate in the above sense with respect
to this semantics. It's clearly always the case that if something's
*true then it's *true that it's *true, and given infinitely many
inaccessibles the converse may easily be seen to hold as well. The liar
paradox is resolved by failure of bivalence.

> 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.

Well, you yourself didn't like the "truth as provability" idea, did
you? Presumably we can all agree on that one. 

> 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".

The main instance of using facts about the modal logics as the basis
for the argument was Richard Heck pointing out that []A->A is not a
theorem of J. Let's just put that fact into context by recalling the
main theorem about J:

"Theorem 2 (Solovay). Let A be a modal sentence. Then (A), (B), and (C)
are equivalent:
(A) For all *, ZF proves A*
(B) A is valid in all finite strict linear orderings
(C) J proves A." 

(Said "satisfies" last time, should have said "proves").

Now, when we have in mind the equivalence between (A) and (C) we might
say "Oh well, the fact that []A->A is not a theorem of J just reveals
something about the limitations of ZF." When we have in mind the
equivalence between (B) and (C), we might reasonably claim that it's
revealing something about a broad class of notions of "necessary
truth", of which your notion happens to be one. But you might
reasonably claim that this is just getting off the subject.

But the fact that you can use a modal logic (and Kripke's semantics for
modal logic) to study the notion at all to my mind does give some basis
for saying that it is in some respects at least a bit like a "notion of
necessary truth", and if the notion is actually the notion of
provability in such-and-such a formal system I would say it's fair to
call it first and foremost a "notion of provability". 

The question is whether it is also reasonable to call it a notion of

> 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.
> 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]
> 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).
> Is TRUTH (in set theory) DEFINABLE (in set theory)?

Yes, given infinitely many inaccessibles. 

Let F(x) be "The sentence with Goedel code x is TRUE". Suppose we can
prove that if a sentence is TRUE, it's TRUE that it's TRUE, and the
converse. That will suffice to show that TRUTH is DEFINABLE (by showing
that F(x) DEFINES TRUTH). 

It's easy to prove that if it's TRUE that something's TRUE, then that
thing is TRUE.

And, given infinitely many inaccessibles, it's easy to prove that if a
sentence is TRUE, it's TRUE that it's TRUE.

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

Yep, then you get that result as well. 

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

Well, you're certainly getting closer and closer to "ordinary truth",
aren't you? Why not just go there?

Well, it seems to me an argument against this conception would have to
come from a sentence which is clearly true but not TRUE. (TRUTH implies
truth if the universe is Mahlo).

How about the sentence "I am not TRUE"? It's neither TRUE nor FALSE,
and therefore it's true. 

Your argument for adopting this "alternative semantics" must presumably
be partly that you like the first way of putting it better.

If kappa is the alpha'th inaccessible cardinal, then if alpha is a
limit ordinal, V_kappa satisfies the sentence "I am not TRUE", but if
alpha is a successor ordinal, it doesn't. 

How about "TRUE*", which means "true in V_kappa with some kappa which
the alpha'th inaccessible with alpha a limit ordinal, and true in all
subsquent V_kappa with this property"? Or how about "TRUE**", which
means "true in V_kappa with some kappa which is an omega-huge cardinal,
and true in all subsequent such V_kappa?" (skipping a few steps here).

It seems to me you will inevitably get pushed closer and closer to
ordinary truth. I'd just go there.

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