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Formally Unknowability, or absolute Undecidability, of certain arithmetic formulas.
Posted:
Jan 27, 2013 12:22 PM


In some past threads we've talked about the formula cGC which would stand for:
"There are infinitely many counter examples of the Goldbach Conjecture".
Whether or not one can really prove it, the formula has been at least intuitively associated with a mathematical unknowability: it's impossible to know its truth value (and that of its negation ~cGC) in the natural numbers.
The difficulty to prove such unknowability, impossibility, is that there are statements that are similar in formulation but yet are known to be true or false. An example of such is:
"There are infinitely many (even) numbers that are NOT counter examples of the Goldbach Conjecture".
The difficulty lies in the fact that there have been no formal logical way to differentiate the 2 kinds of statements, vizaviz, the unknowability, impossibility.
In this thread, we propose a solution to this differentiation difficulty: semantic _reinterpretation_ of _logical symbols_ .
For example, we could reinterpret the symbol 'Ax' as the Specifier (as opposed to Quantifier) "This x", and 'Ex' as the Specifier "That x". And if, for a formula F written in L(PA) (or the language of arithmetic), there can be 2 different "structures" under the reinterpretations in one of which F is true and the other F is false, then we could say we can prove the impossibility of the truth value of F as an arithmetic formula in the canonical interpretation of the logical symbols 'Ax' and 'Ex'.
(Obviously under this reinterpretation what we'd mean as a language "structure" would be different than a canonical "structure").
Again, this is just a proposed solution, and "This x" or "That x" would be not the only choice of semantic reinterpretation. As long as the semantic reinterpretation makes sense, logically at least, it could be used in the solution.
But any constructive dialog on the matter would be welcomed and appreciated, it goes without saying.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 



