Re: Yet another inane amateur Godel question


Arturo Magidin <magidin@xxxxxxxxxxxxxx> writes:

No; the system produces a statement which is *purely about numbers*
and relations between the numbers. It is at the meta-language level
that one can *interpret* the statement as making an assertion about
provability;

And similarly the formalisation of the theorem "x^n * x^m = x^(n+m)"
in the language of arithmetic says nothing about exponentiation; it is
only at the meta-language level that one can *interpret* the statement
as making an assertion about exponentiation?

It is in fact a good idea to forget all about Gödel sentences. The
first-incompleteness theorem is better explained, at this level of
generality, as follows. By the MRDP theorem whether a Diophantine
equation P(x1, ..., xn) = 0 has any solutions is recursively
unsolvable. Hence by standard recursion theoretic considerations, for
any consistent formal theory T of arithmetic there are infinitely many
Diophantine equations P(x1, ..., xn) = 0 which have no solutions but
for which "P(x1, ..., xn) = 0 has no solutions" is undecidable in
T. This explanation should dispel any idea that incompleteness is
about strange self-referential statements etc.

(We need the Gödel sentence only when we get to the proof of the
second incompleteness theorem.)

the Goedel statement, on its face, however, is making an objective
statement about numbers and properties of numbers.

Is there something less objective about statements about finite
sequences (or finite trees) of formulas and properties of such finite
sequences?

And the point is to show that there *are* statements that are
formally undecidable; not that there are "interesting" or "relevant"
statements that are undecidable.

Why? If incompleteness was restricted to some class of weird and
mathematically irrelevant statements surely that would be highly
significant; and surely that the incompleteness theorem establishes
incompleteness for the class of Pi-1 statements -- which includes
Fermat's last theorem, the Goldbach conjecture, Riemann hypothesis,
.... -- is very much to the point.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxx)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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