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Posts:
796
Registered:
9/1/10
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Re: Formally Unknowability, or absolute Undecidability, of certain arithmetic formulas.
Posted:
Jan 28, 2013 1:22 AM
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> 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
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