```Date: Jan 27, 2013 12:22 PM
Author: namducnguyen
Subject: Formally Unknowability, or absolute Undecidability, of certain arithmetic<br> formulas.

In some past threads we've talked about the formula cGCwhich 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 leastintuitively associated with a mathematical unknowability: it'simpossible to know its truth value (and that of its negation ~cGC) inthe natural numbers.The difficulty to prove such unknowability, impossibility, is thatthere are statements that are similar in formulation but yet areknown 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 formallogical way to differentiate the 2 kinds of statements, viz-a-viz,the unknowability, impossibility.In this thread, we propose a solution to this differentiationdifficulty: semantic _re-interpretation_ of _logical symbols_ .For example, we could re-interpret the symbol 'Ax' as theSpecifier (as opposed to Quantifier) "This x", and 'Ex' asthe 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 trueand the other F is false, then we could say we can provethe impossibility of the truth value of F as an arithmeticformula in the canonical interpretation of the logicalsymbols '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, logicallyat least, it could be used in the solution.But any constructive dialog on the matter would be welcomed andappreciated, it goes without saying.-- ----------------------------------------------------There is no remainder in the mathematics of infinity.                                       NYOGEN SENZAKI----------------------------------------------------
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