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The Absolute Arithmetic and Geometric Continua

Published online by Cambridge University Press:  28 February 2022

Philip Ehrlich
Philip Ehrlich*
Affiliation:
Brown University
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Extract

With the appearance of J.H. Conway's On Numbers and Games (1976), the mathematical and philosophical communities have much to celebrate. It is Conway's important discovery that the familiar Dedekind cut and von Neumann ordinal constructions are part of a more general construction which leads to a proper class of numbers embracing the reals and the ordinals as well as many less familiar numbers including -ω, ω/2, l/ω, √ω and ω-π, where ω is the least infinite ordinal. Conway further shows that the arithmetic of the reals may be extended to the entire class yielding a real-closed ordered field (1976, pp. 40-42); that is, an ordered field where every positive element is a square, and every polynomial of odd degree with coefficients in the field has a solution in the field.

Type
Part VII. Mathematics
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1986 , Issue 2: Volume Two: Symposia and Invited Papers 1986 , 1986 , pp. 237 - 246
Copyright
Copyright © 1987 by the Philosophy of Science Association

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Footnotes

1

This essay is an expansion of a portion of the material presented by the author at the meetings in Pittsburgh. Many of the results reported at that time have been deleted and will be discussed in an expanded version of the original paper being prepared for a forthcoming issue of Synthese concerned with theories of continua. Proofs of a few of the deleted results can be found in Ailing and Ehrlich's (1986a and 1986b).

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