With a such amount of knowledge available to current science, its helpless state seems as something irrational and even unthinkable. Nevertheless, it permeates whole of it through and far from only the FLT problem, but also in general wherever you poke, the same thing happens everywhere – science shows its inconsistency so often and in so many questions that they simply cannot be counted. The only difference is that some of them still find their solution, but with the FLT science has been stuck for centuries. However, the greatness of this problem lies in the fact that it, apart from purely methodological difficulties, points to some aspects of a fundamental nature, which have such a powerful potential that, if it succeeds in uncovering of it, science will be able to make an unprecedented breakthrough in its development.
Fermat paid attention to this aspect and was the first to notice even then, that science had no roots to support it as a whole. Simply put, the logical constructions used in solving specific problems do not have a solid support that determines the way, in which each separate branch of knowledge exists. If there is no such support, then science has no protection from the appearance of all kinds of ghosts taken as real entities. The Basic or as it is also called Fundamental Theorem of arithmetic is a vivid for it example. It would seem, what is simpler, one needs only to accept as an unchangeable rule that the numbers can be either natural ones or derived from them. Anything that does not obey this rule cannot be a number. Given that arithmetic is the only science that no other science can do without, it can be stated that all science cannot do without BTA at all! But science itself is not even aware of the fact that BTA is still not proven. And how do you think why? … This is because science simply does not know what is a number!!!
Even to people far from science, this obvious fact can make a shocking impression. Then the question obviously arises: if science does not know even this, then what can it generally know? In this book we’ll explain what the difficulty is here and suggest a solution to this problem. This immediately draws the need for axioms and basic properties of numbers, which were also previously known, but in a very different understanding. After the definition the notion of number and axiomatics, proof of the BTA is required, since otherwise, most of the other theorems simply could not be proven.