[FOM] Godel, Wittgenstein etc.

[Robbie Lindauer]

We have formal languages L(1...n), Godel Statements (G1...n) and "Godel 
Rules" D(1...n).  Godel rules are those procedures used to produce 
Godel statements.

We have a language (L1).

Godel says that there is a statement not provable in L1 which is 
nevertheless provable by L1's existence, the diagonal statement - G1.

We define a second language (L2) which also has an infinite number of 
provable WFF's.

We add all the rules from L1 plus the rule -

(D1) "Add the diagonal statement."

as an axiom.

To see how that would go:

FTFT
TFTF
TFFT
TFTF

Therefore:

FFFF

By (D1)

Then we can add rules for each of the Godel-Like procedures.  These 
will be (D2...Dn).

Godel says "Hey, you're cheating."  We say, "You started it."  If he 
can go on adding statements, we can go on adding rules.  The (D) series 
of rules is merely infinite in its consequences and is mechanical (a 
computer could begin and continue the process - but perhaps not finish 
it.)  If they can not be mechanically specified, we object to them as 
unclear - what can you mean that you can prove that there are true 
statements in (Ln) using Dn if Dn is not mechanical?

All Godel has proved is that for languages that don't have the D-series 
of axioms (every (D1-proved) statement (G1) is true in Ln), they are 
incomplete.
Alternatively, Godel has proved (a few of) the D-Series!

It rests with Godel to prove that the D-Series is indeed infinite.  
Merely having an infinite number of G-Statements is not sufficient to 
cause problems for the axiomatization of logic and mathematics.

If this could be proved, that there are an infinite number of ways of 
producing G-Statements, then a conjecture, that proof will be 
mechanical and will represent a second series of D-Like axioms, and 
will itself have a mechanism for proving its members.

The only problem in doing this is that we may be thought to be required 
to accept some contradictions in our logic (Wittgenstein wouldn't have 
minded this, I do).

But we aren't.  The liar paradox can be solved directly as can others.  
"This statement is false" is provable in some language Ln, but not 
necessarily in L2.

Our goal in logic is to find that L(n) which has all of the necessary 
Godel-Rules as well as all of the true statements of pure logic.

Best,

Robbie Lindauer