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Analysis of an exponential equation with ordinal variables


Author: J. L. Hickman
Journal: Proc. Amer. Math. Soc. 61 (1976), 105-111
MSC: Primary 04A10
DOI: https://doi.org/10.1090/S0002-9939-1976-0450069-X
MathSciNet review: 0450069
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Abstract: This paper is concerned with the analysis of the equation $ {x^y} = {y^z}$, where $ x,y,z$ are variables ranging over ordinals, and where both sides of the equation are transfinite in value. The method used for this analysis consists in regarding $ y$ as a parameter and $ x$ as an independent variable, and determining necessary and sufficient conditions to be placed upon $ x$ so that the resulting equation in $ z$ has a solution. Extensive use is made of normal form, as well as results in ordinal arithmetic by both Bachmann and Sherman.


References [Enhancements On Off] (What's this?)

  • [1] H. Bachmann, Transfinite Zahlen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Heft 1, Springer-Verlag, Berlin, 1955. MR 17, 134. MR 0071481 (17:134d)
  • [2] Ph. W. Carruth, Roots and factors of ordinals, Proc. Amer. Math. Soc. 1 (1950), 470-480. MR 12, 166. MR 0036807 (12:166a)
  • [3] S. Sherman, Some new properties of transfinite ordinals, Bull. Amer. Math. Soc. 47 (1941), 111-116. MR 2, 255. MR 0003688 (2:255j)
  • [4] W. Sierpiński, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., vol. 34, PWN, Warsaw, 1965. MR 33 #2549.

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DOI: https://doi.org/10.1090/S0002-9939-1976-0450069-X
Article copyright: © Copyright 1976 American Mathematical Society

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