Re: Godel Incompleteness Theorem
- From: LauLuna <laureanoluna@xxxxxxxx>
- Date: Tue, 19 Aug 2008 09:30:09 -0700 (PDT)
On Jul 21, 4:05 pm, thirdmer...@xxxxxxxxxxx wrote:
I recently finshed reading a book about Godel's Incompleteness
Theorem, called the Shackles of Conviction by James R Meyer and I was
knocked sideways by it. although it is a novel, it explains Godel's
proof better than any other explanation I have ever seen. But the
astonishing thing is that the book also pinpoints exactly where there
is a flaw in the proof.
Yes, like you, I thought that Meyer had to be wrong. So I looked at
his websitewww.jamesrmeyer.comand found a fully technical paper on
Godel's theorem. I couldn't see anything wrong with Meyer's paper and
I have completly changed my opinion on Godel's proof. Meyer's stuff is
not the ramblings of some freak - he really knows Godel's proof inside
out.
Meyer says that no-one has been able to find an error in his paper. I
showed it to a couple of friends and they couldn't see anything wrong
with Meyer's argument either. So is there anyone there who can find
anything wrong with Meyer's argument? And if no-one can find anything
wrong with Meyer's argument, doesn't that mean that he is right and
Godel was wrong?
Below I copy a post of mine in sci-logic at
http://groups.google.com/group/sci.logic/browse_frm/thread/3fd9e2fe7b924c74/270a6b8f731207cf?hl=en#270a6b8f731207cf
If you read it attentively you'll see Meyer's argument makes no sense.
I can tell I recently had an email exchange with Meyer which he
finally interrupted without answering the crucial quarions I has
posed.
----
I've come across James R. Meyer's website and taken a look at his
argument at http://jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf
Basically he claims there is a confusion between meta-language and
object-language in the statement of Gödel's theorem V in the 1931
paper:
"For every recursive relation R(x1, ..., xn) there is an n-ary
RELATION SIGN r (with FREE VARIABLES u1, ..., un) such that for all
numbers x1, ..., xn we have:
R(x1, ..., xn) -> Bew(Sb(u1, ..., un, r, Z(x1), ..., Z(xn)))
~R(x1, ..., xn) -> Neg(Bew(Sb(u1, ..., un, r, Z(x1), ..., Z(xn))))"
Meyer claims that Gödel refers to some object-language in which
recursive relations are expressed, so that 'x1, ..., xn' are to be
variables in the meta-language (in 'for all numbers x1, ..., xn') and
also in that purported object-language (in 'R(x1, ..., xn)'). He
claims that the purported confusion invalidates the theorem.
I've argued with him that Gödel doesn't refer to expressions of an
object-language in which recursive relations would be expressed, that
Gödel is actually referring to recursive relations themselves; that
there is no meta- and object-language in the theorem but only
ordinary
English (German) extended with mathematical notation; that Gödel is
USING the expression 'R(x1, ..., xn)' as a variable for n-ary
recursive relations, not MENTIONING it.
I have even constructed some versions in which such an object-
language
actually appears, in order to show Meyer that the theorem can be
clearly stated even if made about an object-language able to express
all recursive relations.
As I see it, Meyer's claim amounts to contending that statements
like:
"For all constant functions f and all numbers x, y:
f(x) = f(y)"
are ill-formed, which is absurd.
Can you see any point in Meyer's contention?
------
Regards
.
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