Alternative analysis: first steps on the numbers

Any new result in the theory of injective mappings on the set of natural numbers is of great importance for an analysis. Such outcomes have been contained in this article which also includes the author´s results that form the principles of alternate analysis. At the first (3/sup nd/ item) the new properties of numerical series have been obtained; in particular, the independence of convergence of an alternating series from permutation of the terms of this series has been proved. It has required (2/sup st/ item) both to split all injective mappings /spl phi/: N /spl rarr/ N into three intersected classes and to prove criterions of surjectivity of the injective mappings /spl phi/: N /spl rarr/ N. The main result of the (5/sup th/ item comprises the follows theorem: If sets A and B are some proper subsets of set C then any injection /spl Phi/ : A /spl rarr/ B can be continued up to bijection /spl phi/ : C /spl rarr/ C). The Famous First Hilbert´s Problem couldn´t have took place, as it follows from this article.