Notre Dame Journal of Formal Logic

Another Characterization of Alephs: Decompositions of Hyperspace

John C. Simms


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.

Article information

Notre Dame J. Formal Logic, Volume 38, Number 1 (1997), 19-36.

First available in Project Euclid: 12 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]
Secondary: 03E05: Other combinatorial set theory 03E10: Ordinal and cardinal numbers


Simms, John C. Another Characterization of Alephs: Decompositions of Hyperspace. Notre Dame J. Formal Logic 38 (1997), no. 1, 19--36. doi:10.1305/ndjfl/1039700694.

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