Perron's paradox, emphasizes the danger of assuming a solution to a mathematical problem exists, if the solution is actually nonexistent.
For example, if we assume the largest natural number exists and it is N, therefore if N>1, then N^2>N, and this contradicts our hypothesis that N is the largest natural number, hence the largest natural number is N=1, and this is illogical, hence it emphasizes the danger of assuming a solution exists while it is actually nonexistent.
I think there is a glitch under its underpinning. Let's ponder it again:
If you do not know what is the largest natural number, or you do not know the largest natural number does not exist, then you have no basis for your mathematical operations, and you have no insight on anything in the naïve mathematics. You would not also understand the sign lesser or greater (< or >), hence you cannot conclude N^2>N, is a paradox (contradiction), because you have no insight on foundation of mathematics, it means you do not know natural numbers (alphabet of mathematics) and the sign > or < is meaningless to you and you cannot conclude N^2>N, is a contradiction because you cannot interpret the inequality N^2>N when you do not understand the comparison sign (> or
It is not the case, in my opinion, that "Perron's paradox, emphasizes the danger of assuming a solution to a mathematical problem exists". What I think that it points out is that not every statement defines an object. So (if we assume the usual ordering on the natural numbers) the statement "the smallest common multiple of 6 and 10" defines a unique natural number while the statement "the greatest natural number does not".
Of course, when we do mathematics we are interested in working with well-defined objects. If not, we would be talking about nothing and we would eventually reach a contradiction. In this regards, Perron's paradox just points out this fact in a very sharp way.
The moral of the story is, as I said, that we always have to prove the existence of the objects we are working with. Discussing what "to exist" means is another very different much more difficult matter.
I agree that it is not a hoax. The strange conclusion comes from assuming that there is a largest natural number. The point of the 'paradox' is that we shouldn't simply assume that something exists. It is important to investigate whether or not it exists.
Dear professors Antonio M. Oller-Marcén and David A. Towers
Thank you for the comments. Regards.
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