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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).

References

Ackermann, W. (1937) “Zur Axiomatik der Mengenlehre.” Mathematische Annalen 131: 336-345.CrossRefGoogle Scholar
Ackermann, W. (1937) “Zur Axiomatik der Mengenlehre.” Mathematische Annalen 131: 336-345.CrossRefGoogle Scholar
Alling, N. and Ehrlich, P. (1986a) “An Alternative Construction of Conway's Surreal Numbers.” Mathematical Reports of the Academy of Science Canada, VIII: 241-246.Google Scholar
Alling, N. and Ehrlich, P. (1986b) “An Abstract Characterization of a Full Class of Surreal Numbers.” Mathematical Reports of the Academy of Science Canada, Vm: 303-308.Google Scholar
Conway, J.H. (1976) On Numbers and Games. London, England: Academic Press.Google Scholar
Ehrlich, P. (forthcoming a). “Absolutely Saturated Models.”Google Scholar
Ehrlich, P. (forthcoming b). “An Alternative Construction of Conway's Ordered Field No.” Algebra Universalis.Google Scholar
Ehrlich, P. (forthcoming c). “Universally Extending Continua.”Google Scholar
Fraenkel, A.A., Bar-Hillel, Y. and Levy, A. (1973) Foundations of Set Theory, Second Revised Edition. New York, N.Y: North-Holland Publishing Company.Google Scholar
Jonsson, B. (1956) “Universal Relational Systems.” Mathematica Scandinavica 4: 193-208.CrossRefGoogle Scholar
Jonsson, B. (1960) “Homogeneous Universal Relational Structures.” Mathematica Scandinavica 8: 137-142.CrossRefGoogle Scholar
Jonsson, B. (1965) “Extensions of Relational Structures.” In The Theory of Models. Proceedings of the 1963 International Symposium at Berkeley. Edited by Addison, J.W., Henkin, Leon and Tarski, Alfred Amsterdam: North-Holland Publishing Company.Google Scholar
Kunen, K. (1980) Set Theory. Amsterdam: North-Holland Publishing Company.Google Scholar
Mendelson, E. (1979) Introduction to Mathematical Logic (Second Edition). New York: D. Van Nostrand Company.Google Scholar
Morley, M. and Vaught, R. (1962) “Homogeneous Universal Models.” Mathematica Scandinavica 11: 37-57.CrossRefGoogle Scholar
Reinhardt, W.N. (1970) “Ackermann's Set Theory Equals ZF.” Annals of Mathematical Logic 2: 189-249.CrossRefGoogle Scholar
Schwabhauser, W. (1965) “On Models of Elementary Elliptic Geometry.” In The Theory of Models. Edited by Addison, J.W., Henkin, Leon and Tarski, Alfred. Amsterdam: North-Holland Publishing Company.Google Scholar
Schwabhauser, W., Szmielew, W. and Tarski, A. (1983) Metamathematische Methoden In der Geometric Berlin: Springer-Verlag.CrossRefGoogle Scholar
Szmielew, W. (1959) “Some Metamathematical Problems Concerning Elementary Hyperbolic Geometry.” In The Axiomatic Method. Edited by Henkin, Leon, Suppes, Patrick and Tarski, Alfred. Amsterdam: North-Holland Publishing Company.Google Scholar
Szmielew, W. (1961) “A New Approach to Hyperbolic Geometry.” Fundamenta Mathematicae 50: 129-158.CrossRefGoogle Scholar
Tarski, A. (1951) A Decision Method for Elementary Algebra and Geometry (Second Edition). Berkeley and Los Angeles. University of California Press.CrossRefGoogle Scholar
Tarski, A. (1959) “What is Elementary Geometry?” In The Axiomatic Method. Edited by Henkin, Leon, Suppes, Patrick and Tarski, Alfred. Amsterdam: North-Holland Publishing CompanyCrossRefGoogle Scholar