Calculus without the concept of limit

Piotr Błaszczyk; Joanna Major

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia (2014)

  • Volume: 6, page 19-40
  • ISSN: 2080-9751

Abstract

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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.

How to cite

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Piotr Błaszczyk, and Joanna Major. "Calculus without the concept of limit." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 6 (2014): 19-40. <http://eudml.org/doc/296293>.

@article{PiotrBłaszczyk2014,
abstract = {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.},
author = {Piotr Błaszczyk, Joanna Major},
journal = {Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia},
keywords = {hyperreals; calculus},
language = {pol},
pages = {19-40},
title = {Calculus without the concept of limit},
url = {http://eudml.org/doc/296293},
volume = {6},
year = {2014},
}

TY - JOUR
AU - Piotr Błaszczyk
AU - Joanna Major
TI - Calculus without the concept of limit
JO - Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
PY - 2014
VL - 6
SP - 19
EP - 40
AB - 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.
LA - pol
KW - hyperreals; calculus
UR - http://eudml.org/doc/296293
ER -

References

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  1. 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. 
  2. Cohen, L. C., Ehrlich, G.: 1963, The Structure of the Real Number System, Van Nostrand Co., Toronto-New York-London. 
  3. Cohen, P. M.: 1991, Algebra, Vol. III, John Wiley & Sons, Chichester. 
  4. Dedekind, R.: 1872, Stetigkeit und irrationale Zahlen, Van Nostrand Co., Princeton, New Jersey. 
  5. Deledicq, A.: 1995, Teaching with infinitesimals, in: F. Diener, M. Diener (ed.), Nonstandard Analysis in Practice, Springer, Berlin, 225-238. 
  6. Goldblatt, R.: 1998, Lectures on the Hyperreals, Springer, New York. 
  7. Hartshorne, R.: 2000, Geometry: Euclid and and Beyond, Springer, New York. 
  8. Kanovei, V., Reeken, M.: 2004, Nonstandard Analysis Axiomatically, Springer, Berlin. 
  9. Keisler, H. J.: 1976, Elementary Calculus: An Approach Using Infinitesimals, Prindle Weber & Schmidt, New York. Revised version http: //www.math.wisc.edu/keisler/. 
  10. 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. 
  11. 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. 
  12. O’Donovan, R.: 2007, Pre-University Analysis, in: I. van der Berg, V. Neves (ed.), The Strenght of Nonstadard Analysis, Springer, Wien, 395-401. 
  13. Robinson, A.: 1966, Non-standard Analysis, North-Holland Publishing Company, Amsterdam. 

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