
THE ANTITHESIS of GODEL'S INCOMPLETENESS THEOREM
Posted:
Nov 20, 2012 12:13 AM


DEFINE a 2 parameter predicate PROOF(THEOREM, DEDUCTIONSEQUENCE)
proof( R , [R] ) < axiom(R) proof( R , [RDED] ) < 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.

[RDED]  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 uncomputable! SCI.LOGIC

