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Re: Formally Unknowability, or absolute Undecidability, of certain arithmetic formulas.
Posted:
Jan 28, 2013 1:22 AM


> 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 >  IAMAHEAOFMYSELF



