[FOM] The liar and the semantics of set theory

Richard Heck heck at fas.harvard.edu
Sun Sep 22 23:08:36 EDT 2002

>Someone...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). Since under this semantics the truths and falsehoods of set theory are r.e., and provable sentences are provably provable, I can't see any problem in defining this notion of set theoretic truth in the set theory which has this truth definition.
That rather depends what one means by "definable". If all you want is a 
formula that is true of exactly the Goedel numbers of the truths, in 
this sense, that is, of the provable sentences, then there obviously is 
such a formula, namely, the one that Goedel shows us how to define. But 
if what you want is something stronger, something that would satisfy 
some version of Tarski's schema T, then there is provably no such formula.

What one would be seeking is a formula T(x) such that, for each sentence 
S, it is true in the relevant sense (that is, true in all models) that:
    T(*S*) <--> S.
But truth in the relevant sense is just provability in ZFC, so the 
non-existence of such a formula follows from Tarski's original theorem.

Can we generalize that? Let S be a set of sentences of the "sufficiently 
strong" language L, and let T be some theory in L. We want to think of S 
as specifying the truths of L. So suppose that S is sound for T, in the 
sense that all the theorems of T are in S. We should suppose further 
that S is closed, in the usual sense, and it ought presumably to be 
consistent. So S is just some closed consistent theory in L. Then the 
question becomes: Could there be a formula of L that defined S-truth in 
T? If defining S-truth just means that all and only sentences in S have 
Goedel numbers in the extension of the formula, then so far, the answer 
may be "yes", as we saw above: S might be a formal theory, indeed, the 
set of theorems of T. But if defining S-truth means that all 
"T-sentences" are in S, then there provably is no such formula.

The proof is via the Liar construction and is basically Tarski's. Let 
B(x) be a formula and suppose that for each sentence A of L we have:
    B(*A*) <--> A
in S. By diagonalization, we have a sentence L such that T proves:
    L <--> ~B(*L*),
and so that is in S. But so is:
    B(*L*) <--> L,
and then, by closure, so is:
    L <--> ~L,
and S is inconsistent.

Whether there is any reasonable condition on S that would entail 
undefinability in the weaker sense is harder to say. If we require that 
S be maximal consistent (that is, complete), that would do: That 
effectively imposes bivalence and so Tarski's original argument applies. 
So what Roger wants is something weaker still. I'd guess that results 
about KF and similar systems would apply here, but am too tired to think 
about it....

G'night all,
Richard Heck

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