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Axiomatic Treatment of Rank in Infinite Sets

Published online by Cambridge University Press:  20 November 2018

R. Rado
R. Rado*
Affiliation:
King's College, London
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1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered by Whitney are finite. In the present note the axiomatic treatment of rank is extended to sets of any cardinal. In the special case of algebraic dependence of elements of a field with respect to a sub-field, similar questions have already been considered by Steinitz [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 4 , 01 August 1949 , pp. 337 - 343
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Rado, R., “A Theorem on Independence Relations,” Quart. J. Math., vol. 13 (1942),83-89.Google Scholar
[2] Steinitz, E., “Algebraische Theorie der Körper,” Crelle 137 (1909).Google Scholar
[3] Whitney, H., “On the Abstract Properties of Linear Dependence,” Amer. J. Math., vol. 57 (1935), 509-533.Google Scholar
[4] Zorn, , “A Remark on Method in Transfinite Algebra,” Bull. Amer. Math. Soc, vol. 41 (1935), 667.Google Scholar