Conway’s field of surreal numbers
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- by Norman L. Alling PDF
- Trans. Amer. Math. Soc. 287 (1985), 365-386 Request permission
Abstract:
Conway introduced the Field ${\mathbf {No}}$ of numbers, which Knuth has called the surreal numbers. ${\mathbf {No}}$ is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff’s ${\eta _\xi }$ condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of ${\mathbf {No}}$. In the process, a tower of subfields, $\xi {\mathbf {No}}$, is defined, each of which is a real-closed subfield of ${\mathbf {No}}$ that is an ${\eta _\xi }$-set. These fields all have Conway partitions. This structure allows the author to prove that every pseudo-convergent sequence in ${\mathbf {No}}$ has a unique limit in ${\mathbf {No}}$.References
- Norman L. Alling, On ordered divisible groups, Trans. Amer. Math. Soc. 94 (1960), 498–514. MR 140595, DOI 10.1090/S0002-9947-1960-0140595-8
- Norman L. Alling, A characterization of Abelian $\eta _{\alpha }$-groups in terms of their natural valuation, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 711–713. MR 175983, DOI 10.1073/pnas.47.5.711
- Norman L. Alling, On the existence of real-closed fields that are $\eta _{\alpha }$-sets of power $\aleph _{\alpha }$, Trans. Amer. Math. Soc. 103 (1962), 341–352. MR 146089, DOI 10.1090/S0002-9947-1962-0146089-X G. Cantor, Bieträge zur Begrundung der transfiniten Mengenlehre. I, Math. Ann. 46 (1895), 481-512. English transl., Dover, New York, 1952.
- Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. MR 0042164, DOI 10.1090/surv/006
- J. H. Conway, On numbers and games, London Mathematical Society Monographs, No. 6, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0450066
- Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379, DOI 10.1007/978-3-642-65505-0
- P. Erdös, L. Gillman, and M. Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. (2) 61 (1955), 542–554. MR 69161, DOI 10.2307/1969812
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2 K. Gödel, The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, Princeton Univ. Press, Princeton, N. J., 1940. H. Hahn, Über die nichtarchimedischen Grössensysteme, Sitz. K. Akad. Wiss. 116 (1907), 601-653. F. Hausdorff, Grundzüge der Mengenlehre, von Veit, Leipzig, 1914.
- Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- D. E. Knuth, Surreal numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1974. MR 0472278 W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1931), 160-196.
- K. Kuratowski and A. Mostowski, Set theory, PWN—Polish Scientific Publishers, Warsaw; North-Holland Publishing Co., Amsterdam, 1968. Translated from the Polish by M. Maczyński. MR 0229526
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234 J. D. Monk, Introduction to set theory, Krieger, Huntington, 1980.
- B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202–252. MR 32593, DOI 10.1090/S0002-9947-1949-0032593-5 J. von Neumann, Zur Einfükrung der transfiniten Zahlen, Acad. Szeged 1 (1923), 199-208. —, Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre, J. Reine Angew. Math. 160 (1929), 227-241.
- Alexander Ostrowski, Untersuchungen zur arthmetischen Theorie der Körper, Math. Z. 39 (1935), no. 1, 269–320 (German). MR 1545505, DOI 10.1007/BF01201361
- O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776, DOI 10.1090/surv/004 W. Sierpiński, Sur une propriété des ensembles ordonnés, Fund. Math. 36 (1949), 56-67. —, Cardinal and ordinal numbers, PWN, Warsaw, 1958.
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249, DOI 10.1007/978-3-662-29244-0
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 287 (1985), 365-386
- MSC: Primary 04A10; Secondary 06A05, 12J15, 12J25
- DOI: https://doi.org/10.1090/S0002-9947-1985-0766225-7
- MathSciNet review: 766225