This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.