Re: "Godel got it all wrong"


Peter_Smith wrote:
I can see how people can get confused about the supposed implications
of Gödel's First Incompleteness Theorem. What fazes me is the
seemingly perennial temptation to say that there is something wrong
with the Theorem itself. (When I was editing one of the philosophy
journals for twelve years, there'd be another "refutation" submitted
every four or five months.) The really odd thing is that the Theorem is
in fact *easy*.

Of course, if you want to prove it Gödel's way (and there's other ways
to choose from), there's work to be done, proving that primitive
recursive functions are representable in Peano Arithmetic (and it
requires a pretty bit of trickery with beta-functions to do that). But
once the spade-work is over, the Theorem is indeed easily demonstrated.
Which is not to belittle Gödel's achievement of course. As Kreisel
says in his memoir, it was in fact very important to Gödel that the
theorem was straightforward, *almost* obvious once one had got one's
head around the philosophical point that truth and provability in this
or that formal system are different concepts.

So why, I wonder, do people still get a bee in their bonnet about
this?! Very puzzling.

But the point, Mr Smith, is that no-one presents Goedels theory. They
present a proof and a conclusion from the proof but that is ENTIRELY
DIFFERENT from what the proof is a proof of. What Godels theorem says,
and what the proof is of are not the same.

Look, there is a circularity here. Godels proof proves what Godels
proof proves. Even the proof is self-referential. The proof does not
set out to prove anything. Godel believes that we can search for proofs
without having a goal or system of searching. Godels proof is not a
proof. It is a mapping, posing as a substantive, integrated
relationship, of a self-consistent system to a self-referencing system.
In other words, the multiplicity of style of a proof is mapped to the
monadic self-referencing object.

.