Date: Nov 20, 2012 12:13 AM
Author: Graham Cooper
Subject: THE ANTITHESIS of GODEL'S INCOMPLETENESS THEOREM
DEFINE a 2 parameter predicate

PROOF(THEOREM, DEDUCTIONSEQUENCE)

proof( R , [R] ) <- axiom(R)

proof( R , [R|DED] ) <- if(L,R) & proof( L , DED )

where if(L,R) matches any inference rule in the formal system.

This is equivalent to MODUS PONENS inference application rule.

theorem(R) <- axiom(R)

theorem(R) <- if(L,R) & theorem(L) #MP

proof() remembers the deductions used by modus ponens in the argument

sequence DED.

----------------------------

[R|DED] - complete deduction sequence right up to theorem R, is a

finite length string, all the terms are from a fixed alphabet or

atleast countable.

The HYPOTHESIS which opposes "G=!proof(G)" being significant for

completeness is

there exists some suitably rich set of axioms such that

for every well formed formula F

exist <t1,t2,t3,,,,F>

or exist <t1,t2,t3,,,~F>

e.g [F | t4 | t3 | t2 | t1 ]

where t1 and possibly other theorems in the sequence are axioms, i.e

given as true.

This would imply the existence of a halting theorem decider.

G. COOPER (BINFTECH)

--

S: if stops(S) gosub S

G. GREENE: this proves stops() must be un-computable!

SCI.LOGIC