Asymptotic Numbers: II. Order Relation, Infinitesimals and Interval Topology

It lias been shown in 181 that the set of asympototic number~ A is a system of genera­ lized numbers i ncl ndin~ isomorphic;ill y the set of real numbers gt, :1s well as the field of formal power (asymptotic) series. In the present paper, w hich is a contin uation of 181, an order relation i n A is i ntroduced due to A t u roin~ out to be a totally-ordered set. The consis tency between the order rel ation and the algebraic operations in A is investi ­ gated and in p:u ticular, i t is shown that the inequalities i n A car. de added and multi­ pl ied as in the set of the real numbers. The notions of in fini tesimals (infinitely small numbers), finite and infinitely large numbers are introduced; A turns out to be a non­ arrhimedean ser. The u~age of inflnitesimals as infinitely l arge numbers along wi th the real numbers is the reason why the terms and the notions introduced in this paper nre very much like those of the non-standard analysis (Robinson's theory of iofinitesimals) [7]*. In connection with the order relation, an interval topology of A is introduced nod some of its properties are established. The theor~ of asymptot ic functions, as well ns the applications to the quantum t heor~, Me put off for a future pnper. The notions or the asymptotic numbers 12. 4, 81 and those of the asymptotic func­ tions [3. 5) are introduced as a subsidiary device for invest igat ion of some problems in quantum theory. For further details about the motivation of thi$ work we ad\'ise the reader to refer to [2, 3, 4. 5. RJ. H111 the knowledge of JRJ IS quite sufficient for the 1111­ dcrst.mding of the present paper.