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Formally Unknowability, or absolute Undecidability, of certain arithmetic
formulas.
Posted:
Jan 27, 2013 12:22 PM
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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.
--
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There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
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