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Topic: THE ANTITHESIS of GODEL'S INCOMPLETENESS THEOREM
Replies: 3   Last Post: Nov 20, 2012 11:15 AM

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Graham Cooper

Posts: 4,348
Registered: 5/20/10
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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