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Topic: Godel Incompleteness Theorem
Replies: 172   Last Post: May 2, 2013 2:49 AM

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 Jack Markan Posts: 6,964 Registered: 2/26/05
Re: Godel Incompleteness Theorem
Posted: Sep 4, 2008 2:07 PM

On Aug 30, 1:55 pm, contact080...@jamesrmeyer.com wrote:

> MoeBlee can talk about semantics as long as he likes,

I've got no agenda to talk about semantics in particular. I'm just

> but if he wants
> to be taken seriously with regard to Gödel?s proof,

Taken seriously by whom? Whether you or anyone takes me seriously has
no bearing on the proof itself. Anyway, nothing I've said about the
proof disqualifies my comments from being taken seriously.

> sooner or later he
> must recognise that is a mathematical proof,

Since I recognized that a long time ago, long before this thread or
the existence of your paper, there's no "sooner or later" of concern
in this regard.

> along with all that that
> entails.
>
> MoeBlee stated
> ?Z(x) = the Godel number of the numeral for x?
>
> Now, when MoeBlee refers to ?Gödel number?, what he is referring to
> the Gödel numbering function (the Phi function) ? a mathematical
> concept.
>
> So what MoeBlee is saying above, in mathemathical terms is:
> ?Z(x) = Phi(the numeral for x)?
>
> Now, I have no idea what mathematical concept MoeBlee intends by ?the
> numeral for x?.

The one that Godel mentions in the paper. For any natural number x,
the numeral for x is the expression 'f...f0' where there are x number
of 'f's.

> But it makes no difference anyway.
>
> Since Z(x) is a number-theoretic relation,

More particularly, Z is a number-theoretic function.

> then the assertion of
> equality of
> Z(x) and Phi(the numeral for x)
> asserts that Phi(the numeral for x) is also a number-theoretic
> relation.

No, that is incorrect.

Z is a number theoretic function, so it takes arguments that are
natural numbers and gives, for each argument, a natural number.

Phi is a function that takes arguments that are are expressions of the
language and gives, for each argument, a natual number.

So for any natural number x, Z(x) is a natural number and so is
Phi(the numeral for x). There is no conflict in that.

(Aside: And it doesn't matter in regard the above comments whether
expressions of the language are considered themselves to be natural
numbers. Whatever they are - sequences of symbols or a natural number
itself - the function phi works on them and returns a value that is a
natural number.)

> But that is impossible, since Phi is defined in terms of
> entities that are not numbers/numerals (the other symbols of the
> formal system).

That is the subject of my aside above, just as Godel mentions the
matter himself that it is not crucial whether we take expressions as
strings of symbols or as numbers. We can choose to commit to either
interpretation for the whole proof, and the arguments still go through
mutatis mutandis.

WHATEVER kinds of arguments Phi takes, the VALUES of the function for
those arguments are natural numbers.

> So MoeBlee?s assertion is an absurdity, in the same way as Gödel?s
> assertion of
> Z(x) = Phi(x) for all x a natural number
> is an absurdity.

No, Godel says Z(x) = Phi(the numeral for x). And there is "absurdity"
there. Again, I'll explain it for you:

Z is a function whose domain is the set of natural numbers and whose
range is a subset of the set of natural numbers.

So, for a natural number x, we have Z(x) is a natural number.

Phi is a function whose domain is the set of expressions (whether
expressions are regarded as strings of symbols or as natural numbers)
and whose range is a subset of the set of natural numbers.

The numeral for a natural number x is the expression 'f...f0' (x
number of 'f's).

So Phi(the numeral for x) is a natural number.

So Z(x) = Phi(the numeral for x) is perfectly sensible.

> And perhaps it was MoeBlee who asserted the strange notion that Z(x)
> is not a number-theoretic relation,

No, it wasn't me.

Once more: Z is function that takes natural numbers as arguments. Z is
merely the composition of the functions "the numeral for" and Phi. For
a natural number x, the numeral for x, gives an expression, then Phi
takes that expression and gives a natural number. Z(x) = Phi(the
numeral for(x)). Both sides of the equation are a natural number. I
don't know why you are having problems understanding this.

MoeBlee

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