On Limits of Monotone Sequences in the AST
DOI:
https://doi.org/10.14258/izvasu(2016)1-19Keywords:
alternative sets theory, hyperreal structures, limit, monotone sequence, exact upper boundaryAbstract
In axiomatics of the alternative set theory (AST), construction of some hyperreal structures is considered rather often. These structures are based on horizons that are initial segments or, in different terms, segments of the class of natural numbers. As a rule, they are wider than the class of all finite natural numbers. Contrary to the classical situation, monotone sequences of elements of such hyperreal structure may have no limits even if they are bounded. Necessary and sufficient conditions for a base segment of the structure when such paradoxical situation is impossible have been obtained already earlier. In this paper, we obtain conditions on the growth rate or decrease rate of monotone sequence with a limit within the given hyperreal structure. The relation between the limit and precise upper and lower boundaries of classes is investigated. Causes and mechanisms of limitlessness are examined. It is shown that harmonic series in a hyperreal structure converge if and only if the base segment of the structure is definable set-theoretically or, in other words, has an exact upper boundary.
DOI 10.14258/izvasu(2016)1-19
Downloads
References
2. Larson, Ron, Bruce H. Edwards. Calculus, 10th ed. — Brooks Cole Cengage Learning. — 2014.
3. Бесов О.В. Лекции по математическому анализу. — М., 2014.
4. Briggs W.L., Cochran L. Calculus. — AddisonWesley, 2010.
5. Хренников А.Ю. Суперанализ. — М., 2014.
6. Stewart J. Calculus: Early Transcendentals. — Brooks Cole Cengage Learning, 2014.
7. Вопенка П. Альтернативная теория множеств. Новый взгляд на бесконечность. — Новосибирск, 2004.
8. Вопенка П. Математика в альтернативной теории множеств. — М., 1983.
9. Repicky M. A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem // Comment. Math. Univ. Carolin. — 2015.
10. Дронов С.В. О свойстве фундаментальности сегментов класса натуральных чисел // Известия Алтайского гос. ун-та. — 2009. — № 1/1.
11. Дронов С.В. О сегментах, сохраняющих предел последовательности // Известия Алтайского гос. ун-та. — 2001. — № 1/1.
12. Козлов С.Д., Дронов С.В. О строении изотонного группоида на классе натуральных чисел в AST // Сиб. матем. журн. — 1994. — № 3.
13. Дронов С.В. К возможности движения π-горизонта в AST. // Известия Алтайского гос. ун-та. — 2005. — № 1.
14. Фихтенгольц Г.М. Курс дифференциального и интегрального исчисления : в 3 т. — М., 2016. — Т. 2.
Downloads
Issue
Section
License
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
