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