Paradox of the interesting numbers

The mathematician Hardy tells that, when it qualified in front of Ramanujan the number 1729 of not very interesting, this one retorted to him that it was more the decomposable whole small number all in all of two cubes in two different ways. However, although used including by the mathematicians, the concept of interesting number is not mathematical. The paradox of the interesting numbers , “shows” that all the integers natural are “interesting”. In fact it emphasizes rather humorous of way impossibility mathematically of defining a relevant concept of interesting number.

The paradox

Let us suppose that one can separate the natural integers in two parts, the first is that of the interesting numbers, the second that of the uninteresting numbers. To suppose that there exists in the second part at least an uninteresting number, smallest of them would consequently become interesting. It thus should be added to the first part. But if there remain uninteresting numbers, smallest of them is in its interesting turn… and it is seen that the process does not finish before to have exhausted all the uninteresting numbers (at the end of a possibly infinite number of stages). There cannot thus be some: all the numbers are interesting.

The “demonstration” rests on the fact that the whole of the natural entireties is ordered well, i.e. any subset not-vacuum of entireties has a smaller element. One can more briefly reformulate it and in a more mathematical form thus. If the unit I of the uninteresting integers is not empty, it has a smaller element which, as a more uninteresting small number, becomes interesting, from where a contradiction. One deduced that I is empty (it is a Raisonnement by the absurdity).

Of course, this “demonstration” has only appearance of it. It does not have any value because the subjective concept of interesting number, is not well defined. However a mathematical demonstration must be formulated in a well specified language. If one tries to take with serious the concept of interesting number, one sees that it is in a certain definite way during the alleged demonstration, i.e. this one comprises a vicious circle, similar to that which one finds, in a more explicit way, in the Paradoxe of Berry.

If one tries to make this proof correct, one leads to a commonplace: to formalize the property of the whole of the interesting numbers used, one must say that complementary to this one does not have a smaller element, which in a well ordered unit is a way, certainly very small little the most complicated, to say that it is empty.

It is noticed that the only property of the entireties used is to form a well ordered unit, but for the probability of the reasoning, it is useful that the order is not arbitrary.

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