[FOM] Godel Sentence

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Vladimir Sazonov says:

  >Yes, but "intuitively true" (according to some kind of intuition) 
  >is not the same as "mathematically true in an absolute sense" 
  >which (the latter) is meaningless.

  The range of mathematically meaningful sentences is indeed
philosophically controversial. However, what is supposed to be the
point of exhibiting a particular mathematical theorem - "there are
true sentences in the language of PA which are not provable in PA" -
as somehow more philosophically doubtful than other mathematical
theorems based on the same principles? In this case, say, the theory
ACA. Any misgivings that people may have about mathematical axioms or
methods of proof are better expressed in terms of an explanation of
what they find doubtful, and why. There is no apparent justification
for harping on particular theorems in mathematical logic as somehow
especially doubtful or philosophically wrongheaded.

  To be sure, as you note, many people make all sorts of more or less
absurd or unjustified assertions with reference to Gödel's theorem. It
tends to overstimulate people's imagination. (Similarly for related
results: see
e.g. http://www.dc.uba.ar/people/profesores/becher/ns.html.) But this
is no reason to abandon mathematical common sense in countering such
absurdities.

  >Yes, but I stressed on a "real" meaning of consistency of PA 
  >which involves only feasible proofs unlike formal arithmetical 
  >sentence to argue that the latter is (intuitively) much more 
  >strong, and "believing" in it requires some additional extrapolation. 

  Your interpretation of sentences of the form "For every n,...", "For
every proof in PA,..." and so on as quantifying over "feasible"
constructions is somewhat familiar to me from your other writings.
In my comment, I only wished to emphasize that there is nothing special
about "PA is consistent" in this context, and that your interpretation
is not in agreement with the ordinary mathematical understanding of
"PA is consistent" or "every natural number has a unique prime
decomposition".

 >Saying that consis(PA) is unprovable 
 >in PA, but true without mentioning (at least implicitly) where 
 >it is true, or according to which kind of intuition, is a 
 >wrong interpretation of Goedel Theorem.

   Again, people may say and mean all sorts of things in connection
with Gödel's theorem, but using the ordinary, mathematically defined,
notion of truth for arithmetical sentences, to say that Con(PA) is
true is exactly the same as saying that PA is consistent. We may of
course raise various questions about how such mathematical statements
are to be understood and justified, but there is no "wrong
interpretation of Gödel's theorem" involved in the observation that
Con(PA) is true (in the ordinary, arithmetically definable sense) but
unprovable in PA, any more than it is a wrong interpretation of logic
or arithmetic to observe that the sentence "every natural number has a
unique prime decomposition" is true and provable in PA. Any extra
metaphysical baggage that you associate with "truth" is irrelevant to
these observations. Any questions you may have about how to understand
or justify these observations are better phrased without using such
notions as "true in a theory", "absolute truth", "intuitively
true". They only serve to muddle the discussion.

  >The question was essentially whether there is some miraculous 
  >mathematical way not based on a derivation in a formalism  
  >of demonstrating a truth of mathematical sentences. The answer 
  >seems trivial and negative. But, nevertheless, there is some 
  >widespread opinion that Goedel Theorem gives a way of obtaining 
  >such sentences and even of demonstrating their truth.

  People say all sorts of things about Gödel's theorem, but it is a
simple mathematical fact that we can prove mathematically the existence
of true sentences in the language of PA which are not provable in PA.
Whether the proof is acceptable or convincing is a different question,
which applies equally to any proof based on the same principles.

  Your comments raise a separate question: the role of formalisms in
mathematical proofs. Formalism is not the beginning of mathematics, or
the heart of its practice, and I don't think your characterization of
mathematical provability is correct. But this has nothing in
particular to do with Gödel's theorem.

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