While studying computer science, during a undergraduate and a graduate career, a course was taken in formal methods. We spoke of algebraic specifications and other formal systems used when describing an application. Means existed for verifying that the specifications were correct. It is the case that such philosophical systems, like mathematics, might be consistent or complete but not both. If, it is consistent. One might prove something as conclusively true or false and be certain of the result. But, one cannot prove somethings. If, it is complete. One can prove any and all things; however, the system will not be consistent. So, some provable theorems would arise that would be contradictory.
Question? Is the set of prime numbers infinite? What is your answer? Who proved it so? Was it you, Clyde? Or, was it Euclid? And, with how many based did Euclid work? Some communities during his era counted by ten. One famous one counted by twenties. And, those digits also served as letters. Plus, a very powerful seafaring community counted by sixties.
Let us consider the number line. It has a pair of extremes, a positive and negative infinity, plus a zero. How many total positions are on that number line? [infinity] + [infinity] + 1. That is for both of the extremes and the locus of zero. So, that total number of loci is the [effective infinity] for that number line.
So, let us pause and consider what [infinity] is using the tenth base. If, we assume that that [infinity] holds the greatest possible digit in base ten, or 9. What would [infinity] be? Most probably, it would be a unending sequence of 9s. Yet, that is the [infinity] at the terminus and not the [effective infinity]. Who could not be represent in the tenth base. It would require a greater one.
And, the [infinity] in that greater base, say hexadecimal, would represent a much larger quantity than that in the lower base. That might be proven based upon the underlying principles of the positional numbering system. And, as a result, the [effective infinity] in that larger base would be much greater than the one in the smaller base and larger than its [infinity].
What would the [infinity] be in hexadecimal? Some might say that it would be an extension of the semester end grade reports of all of the modern mathematicians in history. If, their proof-work was regraded based upon the results which could be derived from this construction.
Answer: It would be an endless and unbounded sequence of Fs.
One common construction seen while studying mathematics was this. One could place any infinite set in a injective correspondence with the positive integers. This includes the evens and the primes; however, these sets grow at different rates. So if, we fixed a location like ninety-nine. One hundred integers exist between zero and ninety-nine, inclusive of the endpoints. How many are even numbers? Fewer than one hundred, correct? What about primes? Fewer still, right-o?
Let us pause and review. What does this tell us about these sets for the fixed locus of a positive [infinity] in any base? What about its associated [effective infinity]?
So, one might and must conclude that the set of primes at [infinity] or the [effective infinity] has a much smaller cardinality than the set of integers. And, that would be for any base by construction, including an infinite one. What does this mean? Is you wrong, Clyde? Or, be that Euclid's error? Well, that be you for you!
Seeing that, a sermon could be given that is consistent with Judeo-Christian and Islamic teachings concerning the folly in idolizing any man. It is best that this passage be abbreviated here. Why? Knowledge puffeth up and divides; while, charity edifieth. And, that word charity is properly translated as love. So, please show other enough love that you do not idolize them. For, HE said, "I will have not any other gods before ME." And, idols must be thrown down. So, HE receives the praise and adoration that HE deserves.
So, mathematicians and any other professionals, especially computer scientists, who are the new kids on the block, take the accomplishments of your field and its great names all with a grain of salt.
And, add some Light while you are at it![Matthew 5:13-20 (KJV)]
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