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Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Posted:
Oct 14, 2012 9:21 PM
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> LHS & (LHS -> (a&b->c)) -> a&b->c
>
> to derive models of backward chainable embedded theories that are
> recursively provable back to the axioms!
>
BINARY MODUS PONENS
a & b & (a&b)->c --> c
This not only WORKS! It derives all proofs!
PROOF = backward chainable binary directed acylic graph
You're still stuck on the 'CLASSIC THEOREM'!
******** 10 YEARS DEBUGGING TO FIX THIS **********
> NOT(PA |- P) & NOT(PA |- ~P).
> That is the classical theorem. Duh.
> [...]
> >> I don't see your point. The theorem says that PA can not decide a
> >> particular formula.
> > no, a particular wff (conditions apply)
> When I say "formula", I mean "WFF". I have no reason to talk about
> non-well-formed formulas.
That's why your proof is wrong.
-------------------------------------
[ERROR 1]
By WFF you mean it has a single reduction tree in predicate calculus
from sub predicates and atomic formula, giving it a unique
interpretation. So you have limited the scope of your proof to boolean
formula (true or false).
-------------------------------
[ERROR 2]
THEN: YOU START ADDING FORMULA AT WILL.
"WE CAN CONSTRUCT ANY FORMULA AND TRY TO ADD IT TO THE THEORY"
ERROR!
PA |- P
This is using an axiom-less or inconsistent theory
CONTRADICTION |- ANYTHING
--------------------------------------
[VALID STEP]
Because P is "true" by some rudimentary reductions (P can't be false)
NOT( PA |- P )
NOT( PA |- ~P )
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[ERROR 3]
P <-> NOT( PA |- P )
so P is TRUE (in PA)
-------------------------------------
[ERROR 4]
P is a WFF in PA
AND
P is TRUE in PA
---> P is a missing theorem of PA
You EQUATE
WFF + TRUE --> THEOREM
-------------------------------------
[ERROR 5]
Because:
(P -> Q) -> (P ^ AXIOM) -> Q
is true for 0 order terms (not formula with quantifiers)
You conclude adding AXIOMS to PA could never filter out the Godel
Statement and call it
MONOTONIC LOGIC!
--------------------------------
Herc
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