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The monotonic sequence theorem and measurement of lengths and areas in axiomatic non-standard hyperrational analysis. / Lovyagin, Yuri N.; Lovyagin, Nikita Yu.
в: Axioms, Том 8, № 2, 42, 01.06.2019.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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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 -
ID: 42349988