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

> ----------------------------------------------------

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