Analysis of an exponential equation with ordinal variables
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- by J. L. Hickman PDF
- Proc. Amer. Math. Soc. 61 (1976), 105-111 Request permission
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
- Heinz Bachmann, Transfinite Zahlen, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 1, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (German). MR 0071481
- Philip W. Carruth, Roots and factors of ordinals, Proc. Amer. Math. Soc. 1 (1950), 470–480. MR 36807, DOI 10.1090/S0002-9939-1950-0036807-0
- Seymour Sherman, Some new properties of transfinite ordinals, Bull. Amer. Math. Soc. 47 (1941), 111–116. MR 3688, DOI 10.1090/S0002-9904-1941-07378-7 W. Sierpiński, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., vol. 34, PWN, Warsaw, 1965. MR 33 #2549.
Additional Information
- © Copyright 1976 American Mathematical Society
- 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