Meyer's Argument against Gödel's Theorem

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?



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Relevant Pages

  • Re: Godel Incompleteness Theorem
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