[FOM] Disproving Godel's explanation of incompleteness
Roger Bishop Jones
rbj01 at rbjones.com
Wed Oct 26 17:07:11 EDT 2005
On Monday 24 October 2005 13:55, praatika at mappi.helsinki.fi
wrote:
> Goedel refers in the relevant passage to Tarski. Tarski
> required that any "materially adequate" truth-definition must
> satify what he called Convention T, that is, it must entail as
> its consequences all instances of the following schema:
> (T) True([p]) <-> p.
>
> He then showed that for any sufficiently rich theory T, there
> is no formula P(x) in the language of T such that T proves
> P([S]) <-> S, for every sentence S of L(T).
>
> The alleged counter-example of Jones, which equates truth with
> provability, is completely irrelevant, for Prov(x) does not
> satisfy Convention T (apply Loeb's Theorem).
The purpose of Tarski's Convention T is to express in the
metalanguage the extensional correctness of a truth predicate.
Since in this case the semantics is stipulative rather than
explicatory, it cannot fail to be extensionally correct, and
Convention T becomes irrelevant.
If one nevertheless attempts to check out convention T and the
attempt yields results which are not all entailed by the
definition then this can only show that the formalisation of the
claim of extensional correctness has been done incorrectly.
If this had been done in full compliance with Tarki's
prescriptions then this would show that his prescriptions were
faulty.
Your statement above which purports to give Convention T is
materially different to that of Tarski, though it is an
understandable misconstrual in this particular case.
Tarski says that on the right hand side of the equivalence there
should be the translation into the metalanguage of the sentence
p. In the case where the object language and the metalanguage
are the same it is forgivable that you should think p its own
proper translation. However, if you examine the results and ask
whether the results in this case do in fact formalise correctly
the claim of extensional correctness then you will find that it
does not. The result in turns out to be false because what it
says is that (the provability of "p" is equivalent to p) in
every model of ZFC. It should assert that the provability of p
is equivalent to (p is true in every model of ZFC), which is of
course true.
A better rendering of the correct implementation of Tarski's
convention in this case would be:
True([p]) <-> a translation of "|= p"
Which is, as it should be, highly uninformative, and immune to
Lob's theorem.
If you wish to invoke Tarski against my semantics, then of course
you can, for on the page before he presents his "Convention T"
he explicitly rejects a provability semantics.
It is understandable that Tarski should in 1930-33 have been keen
to make the distinction between provability and truth, but his
argument here (such as it is) is completely without merit.
This passage however serves to underline that this work is
exclusively concerned with Boolean truth valuations.
Roger Jones
More information about the FOM
mailing list