Abstract
In consequent extension of Cantor’s theory of ordinal numbers 0, 1, 2, …ω,ω + 1, ω + 2, …ω・2, ω・2+1, ω・2+2, …… ω2, ……… , non-Archimedean number models for the real axis have been constructed by Sikorski [9] and, independently, Klaua [4, 5]. Reference [9] introduced integral and rational ordinal numbers; references [4, 5] introduced integral, rational, and real ordinal numbers. The purpose of these constructions is to extend the real axis into the transfinite, at the same time as refining it infinitesimally.
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References
Cantini, A.: 1974, Ph. D. Thesis, Florence.
Cantini, A.: 1979, ‘On non-Archimedean Structures of D. Klaua’, Math. Nachr., 87, 35–41.
Cohen, L. and Goffman, C.: 1949, ‘A theory of transfinite convergence’, Trans. Amer. Math. Soc., 66, 65–74.
Klaua, D.: 1959/60, ‘Transfinite reelle Zahlenräume’, Wiss. Zeitschr. der HumboldtUniversität, Berlin, Math.-Nat. R., IV, 169–172.
Klaua, D.: 1960, ‘Zur Struktur der reellen Ordinalzahlen’, Zeitschr. f. math. Logik und Grundlagen der Math., 6, 279–302.
Klaua, D.: 1961, ‘Konstruktion ganzer, rationaler und reeller Ordinalzahlen und die diskontinuierliche Struktur der transfiniten reellen Zahlenräume, Berlin.
Klaua, D.: 1980, ‘Interval components of non-Archimedean number systems’, in Karl L. E. Nickel (Ed.), Interval Mathematics (pp. 129–143), Academic Press Inc.
Klaua, D.: 1981, ‘Inhomogene Operationen reeller Ordinalzahlen’, Archiv f. math. Logik und Grundlagenforschung, 21, 149–167.
Sikorski, R.: 1948, 1950, ‘On an ordered algebraic field’, Soc. Sci. Litt. Varsovic, Sci. Math. Phys., 41, 69–96.
Zakon, E.: 1969, ‘Remarks on the nonstandard real axis’, in: Applications of Model Theory to Algebra, Analysis and Probability (pp. 195–227), New York.
Appendix
Klaua, D.: 1979, Mengenlehre, Berlin-New York.
Klaua, D. 1981, ‘Eine axiomatische Mengenlehre mit größtem Universum und Hyperklassen’, Monatshefte für Mathematik, 92, 179–195.
Klaua, D.: 1986, ‘Zur Existenz der Superstrukturen in der Nichtstandard-Analysis’, photostat manuscript.
Klaua, D.: 1987, ‘A-priori Konstruktion der Stufenbeziehung in einer modifizierten ZFC-Mengentheorie’, photostat manuscript.
Klaua, D.: 1987, Exakte Grundlegung der Mathematik, Karlsruhe.
Kühnrich, M.: 1978, ‘Superclasses in a finite extension of Zermelo set theory’, Zeitschr. f math. Logik und Grundlagen der Math., 24, 539–552.
Laugwitz, D.: 1978, Infinitesimalkalkül, Mannheim-Wien-Zürich.
Lévy, A.: 1976, ‘The role of classes in set theory’, in G. H. Müller (Ed.), Sets and Classes (pp. 173–215), Amsterdam-New York-Oxford.
Stroyan, K. D. and Luxemburg, W. A. J.: 1976, Introduction to the theory of Infinitesimals, New York-San Francisco-London.
Zakon, E.: 1974, A New Variant of Non-Standard Analysis, Lecture Notes in Mathematics, Vol. 369, pp. 313–339.
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Klaua, D. (1994). Rational and Real Ordinal Numbers. In: Ehrlich, P. (eds) Real Numbers, Generalizations of the Reals, and Theories of Continua. Synthese Library, vol 242. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8248-3_10
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DOI: https://doi.org/10.1007/978-94-015-8248-3_10
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