An explanation of G. Galilei's paradox and the estimate of quantities of both rational and prime numbers

Let () and () be two natural variables that is and The pair () () is said to be C -pair if which are as the neighboring elements in Further we prove (Theorem 3) pair () () this pair is C -pair. Let () be natural variable with unlimited step that is d >0 -. Theorem 3 implies that the () with unlimited step can be defined only some subset and is any infinite set. That implies following conclusion (Statement 6). Let be a set of all prime numbers p : p If now it is obvious that | |0 Injective mapping with is said to be potentially antysurjective one (Definition III). Let be (Example 2) square n -matrix with k, m contains of positive rational numbers q, with. Everyone will easily believe that, if we shall assume only distinct numbers in depends essentially on values of the function, for example Now we accept = If we assume a hypothesis that limm( n )»0,6, then we have. (Example 3) Let ( A ) be a harmonic series (Example 3). We prove that ( A ) is the convergent series in addition to it converges to any infinite large number, though it is well known, its sum is not limited by any finite number. See, please, [1, 2].