Chapter

Long Decimals

W. D. Hart

in Future Pasts

Published in print September 2001 | ISBN: 9780195139167
Published online November 2003 | e-ISBN: 9780199833214 | DOI: http://dx.doi.org/10.1093/019513916X.003.0018
 Long Decimals

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Real numbers have decimal expansions. If we add to the first order theory of real numbers the supposition that there is a natural number different from, and so greater than, each (standard) natural number, then we get more places for digits, longer decimals, and more reals. When nonzero digits occur only in 10 (to the n)th place for n non-standard, we have an infinitesimal. Descartes assumed there is a one-one order-preserving correspondence between the points on a line and the reals. How might space differ, and could we tell if we needed infinitesimals (and their infinite reciprocals) to impose co-ordinates? In the region consisting of points only finitely far from a given point, we would have a model for all of solid Euclidean geometry except the uniqueness of parallels, and there could be an infinite gold nugget that does not fill space.

Keywords: Descartes; Newton; Carnap; B Dreben; mathematics; number; infinitesimal; non-standard analysis; space

Chapter.  5064 words. 

Subjects: History of Western Philosophy

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