We'll define the formula cGC (called counter-Golbach-Conjecture, for lack of better word) in full details below but it's sufficient to say that:

cGC means "There are infinitely many counter examples of Golbach-Conjecture"

while its negation:

neg(cGC) means "There are zero or finitely many counter examples of Golbach-Conjecture".

At the heart of the problem is the question, say Q, of whether or not it's impossible to know, to model theoretically prove or verify, that cGC is arithmetically true or false (i.e. true or false in the natural numbers).

Obviously, and logically, the answer to Q could only be:

- No, it's not impossible to know, prove, cGC is true or false.

- Yes, it's impossible to know, prove, cGC is true or false.

The intention, the proposed answer to Q, here is the 2nd choice of answer:

- Yes, it's impossible to know (to model theoretically verify or prove) cGC is true or false in the natural numbers. Hence, ditto for its negation neg(cGC).

There are some rational for this Yes-choice (the Impossible-choice), based on the notion that the set N, as the standard model for the language of arithmetic, can only be incompletely constructed, as we perceive the natural numbers collectively as N.

However, the consequence for the Yes-choice answer is that the concept of the natural numbers then would be a relativistic concept: relative to the choice of true or false value for the formula cGC. We can freely choose the concept as one in which cGC is true, or as one in which its negation neg(cGC) is true; but in either way the canonical properties of the perceived "the natural numbers" are unchanged. Hence, a notion of relativity of mathematical truth in general has been found.

However, let's us just temporarily rest here, having introduced what Q is, to fully specify the formula cGC, in the language of arithmetic L(PA).

-----------------------------------> The definition of the formula cGC.

Please see the attached file:

http://garden.irmacs.sfu.ca/files/the_formula_cGC.txt