FROM G. GALILEI´S PARADOX UP TO THE ALTERNATE ***YSIS

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Sukhotin A. Статья в формате PDF 119 KB

Having unclosed paradox that of natural numbers are as much how many their quadrates, G. Galilei bequeathed to be cautious in the handling with infinite amounts: " ...there isn´t the place for a property of an equality, and also greater and smaller value there, where the matter goes about infinity, and are applied only to finite amounts" [1, p. 140-146]. An explanation of this paradox can be obtained with some conditions, which have allowed to divide all injective mappings φ: N→N on four classes: 1) finitely surjective, 2) potentially surjective, 3) potentially antisurjective and 4) are as trivial antisurjective mappings. The following statements are proved, in particular:

Theorem 1. The injections of 3-rd and 4-th classes are not bijections.

Theorem 2. If a mapping φ: N→N is bijection, then the following limit equality is fulfilled: lim (φ(n):n)=1.

Theorem 3. There isn´t a bijection between of natural numbers set N and its proper subset А⊂N.

Theorem 3 can be proved also by means of the mathematical induction method or with the helping of the following statement.

Theorem 4. Let A and B be proper subsets of set N of natural numbers and there is an injection , then this mapping φ can be prolonged up to bijection .

The concept of numerical sequence convergence is generalized as follows:

Definition 1. A numerical sequence (а) will be termed as a properly convergent sequence, if

. (1)

This concept gives the substantiation to existence of infinite hyper-real numbers. In particular, the sequence of the partial sums of a harmonic series satisfies to a condition of Definition 1. It is easy to proof following statement by means (1):

Theorem 5. A set of Cauchy´s sequences includes a subset of unlimited those.

Corollary of Theorem 5. The real numbers set R isn´t a complete space if it doesn´t include a subset of infinite hyper-real numbers.

A completeness axiom will be entered: every properly convergent sequence converges

Theorem 6.

Theorem 4.

The defined more exactly concept of numerical series has allowed to prove and to show on examples both a necessary criterion of the numerical series convergence on the extended numerical direct is also sufficient, and the convergence of an alternating numerical series in R does not depend on a permutation of this series addends [2]. For example, let (А)==А=ln2. The series (В) was obtained [3, p. 316-319] from the series (А) by following "procedure": after everyone p of sequential positive addends of the series (А) was put q of the sequential negative addends of this series. The sequence ( ) of partial sums of series (В) converges to number =ln(2 ). It is shown in the report the sequence ( ) of series (В) residuals converges to number =ln( ). Therefore, A= + .

Reference

Galilei G.. Selected Works: In 2 t. -Moscow: "Science", 1964. Т. 1.-571 p. (In Russian)
Sukhotin A.M. Alternative ***ysis principles: Study.-Tomsk: TPU Press, 2002.-43 p.
Fikhtengolts G. M. Course differential and integral calculus: In 3 t., 3-rd edit.- Moscow: "Science", 1967.-Т. 2.-664 p. (In Russian)

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