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Winter 1997 Another Characterization of Alephs: Decompositions of Hyperspace
John C. Simms
Notre Dame J. Formal Logic 38(1): 19-36 (Winter 1997). DOI: 10.1305/ndjfl/1039700694

Abstract

A theorem of Sierpinski of 1919 characterized the cardinality of the continuum by means of lines in two orthogonal directions in the plane: CH if and only if there is a subset S of the plane such that every horizontal cross-section of S is countable and every vertical cross-section of S is co-countable. A theorem of Sikorski of 1951 characterizes the cardinality of an arbitrary set by means of hyperplanes in orthogonal directions in finite powers of that set. A theorem of Davies of 1962 characterizes the cardinality of the continuum by means of lines in nonorthogonal directions in the plane, which, by another theorem of Davies of 1962, may be generalized to finite-dimensional Euclidean space. The main results of this paper unify these analogous theorems of Sikorski and Davies by characterizing the cardinality of an arbitrary set by means of hyperplanes in nonorthogonal directions in that set.

Citation

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John C. Simms. "Another Characterization of Alephs: Decompositions of Hyperspace." Notre Dame J. Formal Logic 38 (1) 19 - 36, Winter 1997. https://doi.org/10.1305/ndjfl/1039700694

Information

Published: Winter 1997
First available in Project Euclid: 12 December 2002

zbMATH: 0890.04001
MathSciNet: MR1479366
Digital Object Identifier: 10.1305/ndjfl/1039700694

Subjects:
Primary: 03E50
Secondary: 03E05, 03E10

Rights: Copyright © 1997 University of Notre Dame

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Vol.38 • No. 1 • Winter 1997
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