[FOM] Godel Sentence

[Torkel Franzen]

Arnon Avron says:

  >1) How should the general concept of "A Godel sentence for a
  >theory T" be defined in 
  >exact, mathematical terms (so the question whether a given 
  >sentence is a Godel sentence will have a clear-cut answer,
  >even if we might not know that answer).

  If by a Gödel sentence for T we mean a sentence that formalizes
"This sentence is not provable in T", I don't think any 
characterization is to be expected, even if we restrict T to first
order theories in the language of arithmetic, since there are many
ways of formalizing the concepts involved. Can we even give any
mathematical definition of a Euclid sentence for T, which is to say a
sentence that formalizes in T "Every natural number has a unique prime
decomposition"?

  If we don't seek any intensional characterization, we can define a
Gödel sentence for T as a sentence A for which there exists a
standard formalization Con(T) of "T is consistent" such that Con(T)
is equivalent to A in T. Here everything about Con(T) is fixed
(including the Gödel numbering used, which is any standard version)
except the choice of formula used to define the axioms of T, which
is only restricted to be a Sigma-formula. For theories where we have
a canonical definition of their axioms we can fix the Sigma-formula
as well.

  >To describe in similar terms a reduction of a sentence in weak 
  >second-order logic to a first-order sentence which is obviously equivalent
  >to it (and this equivalence can straightforwardly be proved in any natural,
  >though incomplete, system for weak second-order logic) seems to me
  >totally inappropriate!,

  So can you indicate in what sense a Gödel sentence is, but a formalization
of the fundamental theorem of arithmetic is not, "a sentence...that expresses
a certain property of addition and multiplication of natural numbers"?

  I can't extract any such sense from your comments. It's not as though
an ordinary formalization in the language of arithmetic of the FTA,
written out in full, would in the least strike us as "saying that
every natural number has a unique prime decomposition". Further, a
Gödel sentence is obtained through a reduction of a sentence in the
language of syntax to an arithmetical sentence which is obviously
equivalent to it (and this equivalence can straightforwardly be
proved in any natural, though incomplete, system of syntax and arithmetic).

---
Torkel Franzen