[FOM] Disproving Godel's explanation of incompleteness

[Kenny Easwaran]

What Godel believed here is, I suppose, just what Tarski proved.  No
language with capability for self reference can define a bivalent truth
predicate that applies correctly to the entire language.  Kripke's
"Outline of a New Theory of Truth" shows how such a language can define
a three-valued truth predicate for itself.  Hannes Leitgeb has shown
recently how to define a two-valued predicate for a (relatively large)
fragment of such a language.

I seem to recall Enderton's "A Mathematical Introduction to Logic"
proving Godel's theorem by taking Tarski's result that a complete
bivalent truth predicate is undefinable, and Godel's result (which Roger
Bishop Jones uses here) that the provability predicate is definable, and
thus showing that since everything provable is (presumably) true, that
provability is not complete and bivalent.

Kenny Easwaran

Roger Bishop Jones wrote:

>Godel believed that:
>
>(A).  The truth predicate of a language cannot be defined in that
>language.
>
>and
>
>(B).  That (A) explains the incompleteness of arithmetic.
>
>Before I learned these facts I already had an explanation of
>incompleteness which seems to me a better one, so when I came
>across Godel's explanation (not so very long ago) I was
>skeptical about whether it was the "right" or even a "true"
>explanation.
>Only today has it occurred to me that Godel's explanation can be
>disproved.
>
>I propose to sketch here such a disproof, which consists in
>describing a language for which (A) is false, whose
>incompleteness cannot therefore be explained by (A).
>
>The counterexample is:
>
>       ZFC under the "provability semantics"
>
>The "provability semantics" is as follows.
>A sentence of ZFC is true iff it is provable in ZFC, false iff
>its negation is provable in ZFC (and otherwise has no truth
>value).
>
>[Note: that this definition is a definition only of the notion of
>truth (simpliciter) of a sentence of ZFC, and no change is made
>to any other aspect of the semantics of ZFC (particularly not to
>the notion of "true under an interpretation"), or to its
>deductive system.
>It can be presented alternatively as:
>       True (simpliciter) <=> true in every model of ZFC
>
>Note: also that for the purposes of this proof this semantics
>does not need to be "correct" (whatever that might mean), it
>need only be well-defined.  Perhaps it needs to be correct for a
>certain portion of the arithmetic statements in ZFC, which it
>is.
>]
>
>Conjecture:
>The truth predicate for ZFC under the provability semantics
>is definable in ZFC under that semantics.
>Hence (A) is false for ZFC under the provability semantics.
>
>Observation:
>The incompleteness of ZFC, in the sense relevant to Godel's
>theorem on the incompleteness of arithmetic, is a purely
>syntactic property, hence even ZFC under the provability
>semantics is incomplete in this sense (though under the
>provability semantics ZFC is complete in the different sense
>that it proves all true sentences).
>
>Conclusion: (B) is false (as well as (A))
>
>[ a shorter disproof might be: (B) is false, because (A) is
>false, presumably you can't explain anything with a falsehood.
>My use of ZFC is a bit of a distraction here, I think the whole
>thing works if you substitute PA for ZFC, though in that case
>the provability semantics is more obviously "wrong", but still
>not (IMO) wrong enough to invalidate the disproof.
>I could offer a semantics for ZFC which is arguably right on the
>money (not "wrong" at all) which may also be a counterexample]
>
>Though Godel was notoriously precise and painstaking in his work,
>I have not seen any attempt by him to make (A) precise.
>Does anyone know whether he ever tried to make this claim precise
>(or showed any awareness that it might be vague or false)?
>
>For anyone interested, here is my explanation of incompleteness:
>
>It is part of our requirements of a formal deductive system that
>proofs in that system provide checkable tokens of truth, and
>hence that proofhood be decidable and theoremhood
>semi-decidable.
>Arithmetic truth is not decidable (as shown by Turing) but a
>complete (in G's sense) deductive system for a language which
>included arithmetic would yield a decision procedure (as noted
>by Turing), hence there can be none.
>
>Or, more briefly, formalisations of arithmetic must be incomplete
>because the truths of arthmetic are not recursively enumerable.
>
>Proving that this is a good explanation is not so easy.
>
>Roger Jones
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