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.

Of course the theorem is correct. What is incorrect is the system.
People sometimes confuse the two. The result is counterintuitive. In
particular obtaining a formula which is true but unprovable is as good
as contradiction. (BTW, the fact that from P & -P everything follows is
not the main reason why inconsistency is problematic.)

This result (that there are true but unprovable formulas) resulted in
wild speculations how and why our brain can comprehend formulas to be
true based on their meening, but which no machine can prove. There are
of course those who say that "Goedel's formula is true" is equivalent
to "T is consistent", and that there is nothing to be puzzled about.
But whether this attempt to rationalize the problem away is correct is
questionable.

As I said what is wrong is the system. A system where there are true
but unprovable formulas is not acceptable (modulo " 'Goedel's formula
is true' is equivalent to 'T is consistent' .") I am puzzled by the
references to 'PA'. Shouldn't it be called 'PA in classical logic'?
Since there is nothing wrong with Peano's axioms it is probably the
logic that is wrong.

So to answer your question why people are puzzled by Goedel's theorem:
they find true but underivable frormulas counterintuitive. I might add
that it is rather obvious why this is puzzling.

.