[FOM] Disproving Godel's explanation of incompleteness

[Roger Bishop Jones]

On Wednesday 19 October 2005 10:38, praatika at mappi.helsinki.fi 
wrote:
> Roger Bishop Jones <rbj01 at rbjones.com>:
> > 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.
>
> And where exactly Goedel says this ?

See Ketland's response for the reference that actually provoked 
me. I read it on p122 of Wang's "Reflections on Kurt Godel".

> > 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.
>
> So we could as well define e.g.
>
> True (simpliciter) <=> a negation-free formula of L(ZFC) ?
>
> Certainly we are here interested in extensionally correct
> truth-predicate, one that satisfies Convention T.

I don't know what you mean by "extensionally correct" here.

I know no reason why a truth predicate need satisfy any condition 
whatever other than actually being well-defined.

Of course, not many truth predicates yield a language which is 
actually useful, and not many of them define languages in which 
that same truth predicate is definable.

> > 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.
>
> Depends on how exactly one definies "definable". For various
> alternatives, this is trivially true.
>
> In one good sense, it was Goedel who showed this, so it is
> strange ideed to suggest that he did not recognize this.

I'm happy to call it trivial.
I did not intend to suggest that Godel did not recognise this.
I have no idea whether he ever considered the question.

> > 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).
>
> Depends on how one formulates Goedel's theorem. RBJ only
> considers the part that demonstrates that there is an
> independent sentence. Often, however, it is states as follows:
> For every sufficiently rich theory T, there is a *true*
> sentence of L(T) which is unprovable in T. The argument is
> impotent with this formulation.

Presumably Godel was speaking of the incompleteness result which 
he actually published, and that is obviously what I am speaking 
of.  Isn't your point explicitly aknowledged in the very 
paragraph you are commenting on?
As I also observed I have possible counterexamples in richer 
accounts of the semantics of ZFC, for which ZFC is not complete 
in either sense.  (this would only need to be a possible 
counterexample, even if it were needed, for an explanation which 
uses a possibly false premise to explain a definitively 
established fact must surely be a poor explanation)

> > Conclusion: (B) is false (as well as (A))
>
> Nothing RBJ says shows that. This is an equivocation fallacy,
> by redefining "true" as "derivable in ZFC".

There is no equivocation and I have not "redefined true".

Godel has given an explanation of which a premise is a false 
generalisation which involves quantification over all languages 
(and in which it is clear that it it languages with semantics 
which are spoken of).
I am entitled to reject the explanation by refuting the premise, 
which can be done by exhibiting any language (syntax + 
semantics) which lacks the property alleged to be universal.
I have chosen the syntax of ZFC, (for which there is no 
universally accepted semantics, but which I would be entitled to 
completely redefine for this purpose if it suited my argument) 
and the semantics which I have called the "provability 
semantics".  This is all perfectly rigourous and honest.

> > [ a shorter disproof might be: (B) is false, because (A) is
> > false, presumably you can't explain anything with a
> > falsehood.
>
> Wrong again! A is correct, in any standard sense of "true".

Senses of "true" are not at issue here.
You will have to identify the hole in my refutation if you want 
to maintain the truth of A.

> A RBJ-style counterargument: define "truth" as "well-former
> formula", and look, truth is decidable.

This might well also be a counterexample.

My counterexample has the minor advantage that it is actually 
consistent with the views of some people (not including me) 
about which sentences of ZFC should be considered true.

> Note that I am not really arguing for (B), I just wanted to
> point out various questionable steps in RBJ's reasoning.

Keep trying.

I myself am warming to Godel's explanation, which can easily be 
fixed and is then better than the one I offered (though still a 
bit terse).

Roger Jones