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From: Charlie-Boo on 30 May 2010 00:16

There are various ways in which we can prove the 3 fundamental results
of Incompleteness in Logic (Godel 1931, Rosser 1936, Smullyan 1961) in
a simple, almost trivial, proof. The entire proof is very short, but
may be completely dominated by self reference. This is in contrast to
published proofs which are many orders of magnitude longer and more
complex, involving intricate considerations of the exact contents of
various wffs of logic.

Consider the following methods:

1. Appeal to the Theory of Computation

Each pair of the sets of the true, provable and unrefutable sentences
differ in whether they are r.e. or co-r.e. (If the system is
consistent then the universal set if the fourth possibility.)

2. Appeal to Self-referential Paradoxical Statements

“This is not P.” for P being true, provable and unrefutable creates
sentences with differing truth values viz. none, true and false.

How do we generalize these 2 methods into one?

How do we generalize these 2 methods to generate a third method?

C-B
From: Charlie-Boo on 30 May 2010 00:40
On May 30, 12:16 am, Charlie-Boo <shymath...(a)gmail.com> wrote:
> There are various ways in which we can prove the 3 fundamental results
> of Incompleteness in Logic (Godel 1931, Rosser 1936, Smullyan 1961) in
> a simple, almost trivial, proof.  The entire proof is very short, but
> may be completely dominated by self reference.  This is in contrast to
> published proofs which are many orders of magnitude longer and more
> complex, involving intricate considerations of the exact contents of
> various wffs of logic.
>
> Consider the following methods:
>
> 1. Appeal to the Theory of Computation
>
> Each pair of the sets of the true, provable and unrefutable sentences
> differ in whether they are r.e. or co-r.e.  (If the system is
> consistent then the universal set if the fourth possibility.)
>
> 2. Appeal to Self-referential Paradoxical Statements
>
> “This is not P.” for P being true, provable and unrefutable creates
> sentences with differing truth values viz. none, true and false.
>
> How do we generalize these 2 methods into one?
>
> How do we generalize these 2 methods to generate a third method?
>
> C-B

There are other methods but they rely heavily on the use of a
particular language CBL. The basic idea is to generalize the notion
of characterizing a set from the published restricted to 3 or 4
kludges "expressible" and "representable" and "contrarepresentable".
P/Q means we can characterize relation or thing P in base Q where Q
could be the provable ("represent"), or true ("express"), or refutable
("contrarepresent") etc sentences.

E.g. Godel 1931 is, where PR=provable, TW=true:

~PR/TW Unprovability is expressible.
=>
-PR,TW Provability is not equal to truth.

where in general,

P/Q
=>
-P,~Q

because

-~P/P

No system can represent its negation.

is the fundamental Axiom of Incompleteness where all incompleteness
proofs end.

I don't know if people would accept something so abstract as a proof
or not.

C-B
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