The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis

Yuri N Lovyagin,Nikita Y Lovyagin

doi:10.3390/axioms8020042

Abstract

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.

Highlights

The non-standard analysis, offered by Robinson in the 1960s [1], considers mathematical objects from a different point of view than the classic ε–δ analysis
Using the theorem on monotonous bounded sequence, we describe the ability of line segment lengths measurement using any number of a class of infinitely close hyperrational numbers as a length, i.e., modeling finite real numbers using finite hyperrational ones
Let us prove the key result of this paper, which is an analogue of the theorem on monotonous bounded sequence of classic real analysis

Summary

Introduction

The non-standard analysis, offered by Robinson in the 1960s [1], considers mathematical objects from a different point of view than the classic ε–δ analysis. It operates infinitesimal and infinite numbers, which are correspondingly strictly greater than or strictly less than any strictly positive finite number by absolute value, and the relation of infinitely closeness ≈ of two numbers, which means that the difference between these numbers is infinitesimal. The limits are substituted either by considering a sequence member with an infinite number or by taking a value of a function in a point infinitely close to the considered one. Non-standard analysis is widely used in pure and applied mathematics, i.e., in mathematical physics, differential equations [6], and economics [7]

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