[FOM] truth and consistency
torkel at sm.luth.se
Sat May 31 02:22:27 EDT 2003
Martin Davis says:
>I feel rather like that regarding philosophers as I read some recent FOM
>postings on the question of in what sense one can be secure in the belief
>that Gödel's undecidable sentence G is true.
The matter is easily explained.
>I think the basic fact is
>straightforward and acknowledged by all. One is entitled to unequivocally
>assert G only if one is convinced of the consistency of the underlying
This is indeed a consequence of the fact that G and "S is
consistent" are provably equivalent (in a weak theory at that). Gödel
sentences exercise a powerful appeal on the imagination, but in
philosophical discussions there is much to be gained by forgetting
about them altogether.
>If the underlying system is PA (first order number theory), I believe that
>one can claim that the evidence for its consistency is simply overwhelming.
>Even the trivial consistency proof based on the standard model uses far
>less than much well-accepted ordinary mathematics.
The dispute over "thin" vs "fat" as regards the concept of truth
turns on precisely such trivial consistency proofs. The trivial
consistency proof establishes much more than consistency, namely that
every theorem of PA is true (equivalently: true in the standard
model). To speak of the truth or falsity of statements in the language
of elementary arithmetic, it is not necessary to postulate the
existence of the set of natural numbers, and to formalize reasoning in
terms of arithmetical truth and falsity, it is sufficient to add to PA
a truth predicate with suitable (Tarski) axioms, and induction axioms
for the extended language. Jeffrey Ketland and others argue that once
we understand and accept the axioms of PA, we also understand "true
statement of arithmetic" and should also accept the axioms of the
extended theory. Since the extended theory is known to be logically
stronger than PA, this indicates that truth (for arithmetical
sentences) is a substantial notion. This conclusion is contrary to a
doctrine known as "deflationism" with regard to the notion of truth.
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