[FOM] Self-referentiality and Godel sentences
Torkel Franzen
torkel at sm.luth.se
Wed Sep 3 20:50:17 EDT 2003
Arnon Avron says:
>Can the same be said about "This sentence is unprovable in T"? Has this
>sentence for itself any reasonable meaning without formalization??
There are indeed significant differences between Gödel sentences and
formalizations of ordinary statements of mathematics, but what I'm
wondering is why self-reference should be any more problematic, in
this connection, than reference in general. You emphasize that
"a Godel sentence for T does not (and cannot) directly refer to itself -
only to some (term for) a number which serves as its code according to
a very concrete, peculiar choice of Godel numbering." For this reason
you consider it misleading to call a Godel sentence self-referential.
Well, if we think that people are in danger of being misled by
the description of Godel sentences as self-referential, we need
only state explicitly what this means (as has been done in this
thread).
You remark that people say and believe many strange things about
Gödel's theorem, and suggest that describing Godel sentences as
self-referential may encourage misconceptions about the theorem. In my
experience, there is no way of formulating the theorem that forestalls
misconceptions, and we need to explain "self-referential", "true",
"theory", "unprovable" and so on over and over again anyway, to
deal with the various misconceptions.
So, to return to my earlier question: is it not equally true that a
formalization in arithmetic of Ramsey's theorem, or the fundamental
theorem of arithmetic, or any of a host of statements in finite
mathematics, does not and cannot directly refer to finite sets or
sequences of numbers, but only to some numbers which serve
as a code for or representation of such sets or sequences according to
a very concrete, peculiar choice of representation? Does this mean
that the formalizations do not in fact express the statements in
question?
---
Torkel Franzen
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