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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: Aug 22, 2008 12:56 PM

On Aug 22, 2:15 am, contact080...@jamesrmeyer.com wrote:
> On Aug 22, 3:52 am, MoeBlee <jazzm...@hotmail.com> wrote:

> > So, in the instance under discussion, Z(x1) is an expression of the
> > formal system.

>
> No, it's not. That's the whole point of meta-language and sub-
> language. A meta-language can talk about the symbol combinations of
> the sub-language as combinations of symbols.
>
> But once you say that, in a meta-language, that A = BC, you are
> assigning an equality between A and B, you have to at the same time
> IGNORE any physical properties of A and BC. Otherwise you could then
> state that A consists of one symbol - but because you have assigned an
> equality between A and BC, then it would also follow that BC consists
> of one symbol.
>
> So that means that assigning an equality such as Z(x1) = ?some formal
> combination? precludes referring to the physical attributes of either
> side of that equality in the same set of propositions that constitute
> a proof, and that means that yoiu cannot apply that equality at the
> same time as applying the Godel numbering system to ?some formal
> combination?.
>
> I?m afraid that it you that is confused.

I can see that either you didn't read my explanation for you, or you
don't understand it. As well as do have a confusion of use and
mention.

Z is a number-theoretic function whose domain is the set of natural
numbers. For any natural number x, we have Z(x) = the Godel number of
the object-language numeral for x. Also, Godel as much as takes
expressions of the object language to be their Godel numbers, so the
Godel number of an expression of the object language is tantamount to
that expression itself (indeed as Godel says that Z(x) IS the numeral
for the number x). We could just as well insert the inverse of the
operator phi at the appropriate places to get not the Godel numbers
but instead back from the Godel numbers to the expressions themselves.
Understanding this is part of the context of understanding the entire
paper, indeed as Godel discusses the matter early in the paper.

And you skipped again the distinction between Z(x) and 'Z(x)'. First,
though, Z(x) is the Godel number of not a single symbol of the object-
language, but rather of a numeral of the object language, which is
only a single symbol (or, depending on how one would strictly
formalize, a sequence of length 1 made from that single symbol) only
in the case where x is 0; otherwise Z(x) is the Godel number of the
sequence of symbols 'f'...'f''0' (x number of 'f's'). So, again Z(x)
is the Godel number of an expression of the object-language (and as we
may take expressions to be their Godel numbers), and since Z(x) is the
Godel number of 'f....f0' for x number of f's, Z(x) is not the
juxtaposition of 'Z' with '(' with 'x' with ')'. Rather it is 'Z(x)'
that is the juxtaposition of 'Z' with '(' with 'x' with ')'. This is
basic use/mention distinction.

MoeBlee

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