[FOM] Re: Diagonalization and Self-Reference

[Torkel Franzen]

Richard Heck writes:

 >What I'm trying to suggest is that this difference, as unimportant as it 
 >may be for some purposes, does matter, and even can matter for 
 >mathematical (and not just "philosophical") purposes. Both these points 
 >follow, or so it seems to me, from diagonalization's failure to capture 
 >the intuitive inconsistency of (2). That just shows that the technique 
 >of diagonalization in the language of arithmetic does not produce 
 >sentences that are in the strictest sense self-referential.

  I don't understand why you speak here of a "failure of
diagonalization".  If by a self-referential sentence we mean a
sentence of the form A(t), where the value of t is (a Gödel number
for) A(t) itself, then indeed we don't know of any self-referential
sentences in the language of PA. If we also admit sentences of the
form (Ex)(P(x)&Q(x)) where P(x) is true only of the sentence
(Ex)(P(x)&Q(x)) itself, then there are self-referential sentences in
the language of PA.

  A self-referential statement of the form A(t) in an extension by
definitions of PA can be translated into a self-referential statement
A* in the language of PA. Why should we expect A* to have the same
truth value as A(t)? Self-referential statements don't usually
retain their truth value in translation (consider e.g. "This is an
English sentence"). It so happens that the Gödel sentence does retain
its truth value in translation, because it deals only with the
provability of A(t), which is preserved under translation.