[FOM] Disproving Godel's explanation of incompleteness
Roger Bishop Jones
rbj01 at rbjones.com
Tue Oct 18 04:55:58 EDT 2005
Godel believed that:
(A). The truth predicate of a language cannot be defined in that
(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"
Only today has it occurred to me that Godel's explanation can be
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
[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
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
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.
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
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.
More information about the FOM