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Foundations of Science

, Volume 23, Issue 4, pp 681–704 | Cite as

Bolzano’s Infinite Quantities

  • Kateřina Trlifajová
Article
  • 79 Downloads

Abstract

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.

Keywords

Bernard Bolzano Paradoxes of the infinite Measurable numbers Cantor’s Set Theory Infinite quantities Non-standard analysis 

Notes

Acknowledgements

I am grateful to Mikhail Katz, Jan Šebestík and Petr Kůrka for helpful comments on an earlier version of the manuscript. Many thanks to my colleague Jan Starý for fruitfull discussions about ultrafilters and to my husband Jan Trlifaj for notes on the final version. I am also very grateful to the two anonymous referees for their thorough reading of the manuscript and valuable comments.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Information TechnologyPraha 6Czech Republic

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