[FOM] Godel Sentence & The Liar

[Torkel Franzen]

Arnon Avron says:

  >A Godel Sentence
  >is a sentence in the language of elementary arithmetics that expresses
  >a certain property of addition and multiplication of natural numbers.

   You can say as much about any formalization in arithmetic of a
mathematical statement, say one about the rational numbers. The Godel
sentence "says of itself that it is unprovable" in a straightforward
sense, just as the formalization in PA of the fundamental theorem of
arithmetic "says that every natural number has a unique prime
decomposition". A coding is always presupposed in such formalizations.

  >But I doubt that
  >one may find conditions which are general enough to cover any
  >sentence that one might call in certain circumstances
  >"a Godel sentence for the theory T". 

  "The" Godel sentence for T, when we are talking about an
effectively axiomatizable theory T, is equivalent in T to "T is
consistent", where "T is consistent" is formalized as "no contradiction
can be derived from the set of formulas a such that P(a)", where P(a)
is a Sigma-formula (RE-formula). For standard theories T (PA and so on)
there is a canonical P(a) ("a is one of the axioms A1,..,An or
an instance of one of the schemas S1,..,Sm"). For general effectively
axiomatizable T there is no such canonical P(a), and different
choices of P(a) yield different versions of "T is consistent" that
are not equivalent in T. Godel's theorem applies to all these "T is
consistent", however.

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Torkel Franzen