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.

Shy a Doctoral Degree

The very reason why it was decided that earning a doctoral degree in computer science was impractical is the immaturity of the field. It seemed that as learning advanced certain basic aptitudes and understandings were lost.

One of the most glaring was the notion that infinity is not reachable.

This is contradicted by G. Cantor’s diagonalization argument. Which is fine for reasoning about the comparative growth of sets; however, it only truly shows that our positional numbering system has limitations. Which are not found among other numbering systems. Like, those based upon the grouping of quantity symbols. Such as, the system of Roman numerals.

For, it is personally believed that Cantor did not find nor fix infinity at a point. Consider, Cantor’s argument that the largest possible base ten number (999…) is along a diagonal in the table he created. Which is necessarily not square. As, it is often represented. When, this argument is “proven”. Since, the cardinality of a set of numbers in base ten outpaces its length significantly. Intuitively, how many numbers might be held is a set of numbers who are nine digits in length? One billion or ten with an exponent of nine.

Yet, adding any number in the set of all possible numbers representable by base ten and this largest number (999…) will yield a value smaller than the largest possible number found in base thirty-six (ZZZ…). With the alphabet serving as digits for numerals greater than ten.

So, Cantor did not locate a “true” infinity. His argument proves that quantities exist beyond the largest possible base ten value; however, that scalar is simply finite. Which is intuitive since ten is also. So, our base ten positional number system has limits.

This represents a limit of computation for the base but not computability in general.

It is highly possible that such a numbering system supporting a infinite base does not. Seeing that, infinity plus infinity is a new infinity. Yet, for mankind it is often most comfortable discussing computation in terms of base ten.

The popularity of this argument, Cantor’s diagonalization, and others that were seemingly as nonsensical are the very reason why doctoral studies were left. It was hoped that certain fundamental skills and aptitudes would never be lost by practicing very popular and community mandated dysfunctions as a doctoral-level education and career were sought. When, one might show that subtraction is not computable based upon Cantor’s argument. One has reached a sufficient and sound contradiction in his proof-work showing that Cantor’s argument is a fallacy; however, it seems that among modern computer scientists this is only fuel and motivation for more baffling proof-work establishing the hardness and intractability of some problems who were only meant as curriculum-based heuristics.

The words of a very well-respected and widely published professor in the Texas System will never be forgotten. He said, “If, you want your PhD. You will do it this way [and not question it]!” Then, he suggested a book by G. Polya on proper proof-work.

These ideas were shared with Co-Pilot from Microsoft and the system tanked. Seemingly, it feels that integers cannot have a infinite length; yet, if they cannot, then one cannot construct the diagonalization argument of Cantor.

And although, it has been many years since his name was first seen along with his diagonalization proof. It was thought that his first name was Gregor and not Georg. Yet, those are memories arising from the early-80s.