PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | 6 | 19-40
Tytuł artykułu

Calculus without the concept of limit

Treść / Zawartość
Warianty tytułu
Języki publikacji
PL
Abstrakty
PL
There are two different approaches to nonstandard analysis: semantic(model-theoretic) and syntactic (axiomatic). Both of these approachesrequire some knowledge of mathematical logic. We present a method basedon an ultrapower construction which does not require any mathematical logicprerequisites. On the one hand, it is a complementary course to a standardcalculus course. On the other hand, since it relies on a different intuitivebackground, it provides an alternative approach. While in standard analysisan intuition of being close is represented by the notion of limit, in nonstandardanalysis it finds its expression in the relation is infinitely close. Asa result, while standard courses focus on the " − technique, we explorean algebra of infinitesimals. In this paper, we offer a proof of the theoremon the equivalency of limits and infinitesimals, showing that calculus can bedeveloped without the concept of limit.
Słowa kluczowe
PL
Twórcy
Bibliografia
  • Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Sherry, D., Shnider, S.: 2013, Is mathematical history written by the victors?, Notices of The American Mathematical Society 7, 886-904.
  • Cohen, L. C., Ehrlich, G.: 1963, The Structure of the Real Number System, Van Nostrand Co., Toronto-New York-London.
  • Cohen, P. M.: 1991, Algebra, Vol. III, John Wiley & Sons, Chichester.
  • Dedekind, R.: 1872, Stetigkeit und irrationale Zahlen, Van Nostrand Co., Princeton, New Jersey.
  • Deledicq, A.: 1995, Teaching with infinitesimals, in: F. Diener, M. Diener (ed.), Nonstandard Analysis in Practice, Springer, Berlin, 225-238.
  • Goldblatt, R.: 1998, Lectures on the Hyperreals, Springer, New York.
  • Hartshorne, R.: 2000, Geometry: Euclid and and Beyond, Springer, New York.
  • Kanovei, V., Reeken, M.: 2004, Nonstandard Analysis Axiomatically, Springer, Berlin.
  • Keisler, H. J.: 1976, Elementary Calculus: An Approach Using Infinitesimals, Prindle Weber & Schmidt, New York. Revised version http: //www.math.wisc.edu/keisler/.
  • Lindstrøm, T.: 1988, An invitation to nonstandard analysis, in: N. J. Cultland (ed.), Nonstandard analysis and its applications, Vol. 10, Cambridge University Press, Cambridge, 1-105.
  • Łoś, J.: 1955, Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, in: T. Skolem et al. (ed.), Mathematical interpretation of formal systems, North-Holland, Amsterdam, 98-113.
  • O’Donovan, R.: 2007, Pre-University Analysis, in: I. van der Berg, V. Neves (ed.), The Strenght of Nonstadard Analysis, Springer, Wien, 395-401.
  • Robinson, A.: 1966, Non-standard Analysis, North-Holland Publishing Company, Amsterdam.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.ojs-issn-2450-341X-year-2014-volume-6-article-3654
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.