```Date: Jan 28, 2013 1:22 AM
Author:
Subject: Re: Formally Unknowability, or absolute Undecidability, of certain arithmetic formulas.

> 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, viz-a-viz,> the unknowability, impossibility.> > In this thread, we propose a solution to this> differentiation> difficulty: semantic _re-interpretation_ of _logical> symbols_ .> > For example, we could re-interpret 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 re-interpretations 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 re-interpretation 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> re-interpretation.> As long as the semantic re-interpretation 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
```