TY - JOUR

T1 - The monotonic sequence theorem and measurement of lengths and areas in axiomatic non-standard hyperrational analysis

AU - Lovyagin, Yuri N.

AU - Lovyagin, Nikita Yu

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

KW - Axiomatic non-standard analysis

KW - Hyperrational numbers

KW - Line segment measurement

KW - axiomatic non-standard analysis

KW - 26E35

KW - line segment measurement

KW - hyperrational numbers

UR - http://www.scopus.com/inward/record.url?scp=85066846835&partnerID=8YFLogxK

U2 - 10.3390/axioms8020042

DO - 10.3390/axioms8020042

M3 - Article

AN - SCOPUS:85066846835

VL - 8

JO - Axioms

JF - Axioms

SN - 2075-1680

IS - 2

M1 - 42

ER -