Abstract

It has been shown in [8] that the set of asympototic numbers A is a system of generalized numbers including isomorphically the set of real numbers R, as well as the field of formal power (asymptotic) series. In the present paper, which is a continuation of [8], an order relation in A is introduced due to A turning out to be a totally-ordered set. The consistency between the order relation and the algebraic operations in A is investigated and in particular, it is shown that the inequalities in A can be added and multiplied as in the set of the real numbers. The notions of infinitesimals (infinitely small numbers), finite and infinitely large numbers are introduced; A turns out to be a nonarchimedean set. The usage of infinitesimals as infinitely large numbers along with the real numbers is the reason why the terms and the notions introduced in this paper are very much like those of the non-standard analysis (Robinson's theory of infinitesimals) [7]*. In connection with the order relation, an interval topology of A is introduced and some of its properties are established. The theory of asymptotic functions, as well as the applications to the quantum theory, are put off for a future paper.

The notions or the asymptotic numbers [2, 4, 8] and those of the asymptotic functions [3, 5] are introduced as a subsidiary device for investigation of some problems in quantum theory. For further details about the motivation of this work we advise the reader to refer to [2, 3, 4, 5, 8]. But the knowledge of [8] is quite sufficient for the understanding of the present paper.

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URL: https://digitalcommons.calpoly.edu/math_fac/85