From: Jim Balter 
Date: Tue, 26 Nov 1996 08:53:23 -0800
Organization: JQB Enterprises

Geoffrey Norman Watson wrote:

>  A second point:
>  An important aspect of these results is that in some way "provability" is
>  weaker than "truth" (this has the advantage that provability thus escapes
>  from the truth paradoxes) and they explore the differences.  So it is wrong
>  to equate the two by saying "it can't be true unless you can prove it".
>  (Although of course with a consistent system, then we can say "if you can
>   prove it then it is true".)
> 
>  I believe that detailed versions of Goedel's second incompleteness theorem
>  exhibit quite specifically the sentences which are true but which cannot
>  be proved in the relevant system. Their truth is established by the law of
>  the excluded middle and the meta-level relationship between statements
>  of arithmetic and sentences of the theory defined by the Goedel encoding.

Any claim of truth must be justified, and that justification is never
sufficient unless it is a proof.  Anything short of that is subject to
the unanswerable objection that the claim could be wrong.  Of course,
the proof can come from adding axioms.  The whole point is that truth
is relative to a theory.  In order to prove a theory T consistent,
we simply add the axiom "T is consistent" to get T', in which it is
true that T is consistent.
 
>  A theory may have many interpretations, and it may well be that we can
>  recognise truths in some interpretations that cannot be proved in the theory.

We "recognize" this by assuming that the theory is consistent,
and then raising this assumption to a higher psychological status
by calling it "seeing" the truth.

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