Goldbach's Conjecture and Cantor's Argument

 A note was posted yesterday April 27, 26 that described why Cantor's diagonalization argument only describes exceeding the upper boundary set by the largest value that one might represent in base ten; however, that value is only finite and does not represent a true infinity.

Yet, that was only opinion supported with handwaving proof-work. And if, it does not stand the scrutiny of accomplished mathematicians. And, the diagonalization argument concerning infinity holds. Then, one might say that the true "countable" infinity represented by the number line is the cardinality of the integers found on it between (-999...) and (+999...). 

Each of these extreme values have an absolute value of (+999...). And when, they are added. They equal twice (+999...). Which is only representable in a base larger than ten. And, that effective infinity based upon the count of the integers on the number line is even.

Goldbach's Conjecture supposes that every even number is the sum of a pair of primes. Consider, this effective infinity for the base ten number line. It is the sum of composites only (999...) and (999...). Which are each the product of 9 and (111...). And, the largest possible prime smaller than (999...) is (999...) minus 2. And if, this value is prime. The largest even number which is possibly a sum of primes is (999...) minus 2 plus (999...) minus 2. Which is twice (999...) minus 4. So, twice (999...), the effective infinity of the number line, and twice (999...) minus 2, the last even number before this "infinity", are not a composition of primes. These are the minimum pair of even quantities between zero and the effective infinity which are contradictions for Goldbach's Conjecture.

One might argue the contrary that this is the very reason why the infinities identified by Cantor are fallacies. Seeing that, a long-forgotten derivation supporting Goldbach's Conjecture shared that it holds if one can find a singular prime between n and its square root. If so, a sufficient number of primes exists for yielding all of the even numbers less than n. It is thought that such is the case. Although, the nature and details of the proof-work have been long lost over the decades. That was a derivation seen during secondary school calculus and an enrichment module taught by the instructor.

As learnt during Philosophy 100 Logic at a small college in the rural Heartland, one might prove anything in the affirmative if one of his premises is false. Which is a further argument that Cantor's diagonalization argument has its limitations and has been misinterpreted among those who practice the Theory of Computation.