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Distributivity and an axiom of choice

Published online by Cambridge University Press:  12 March 2014

George E. Collins
George E. Collins*
Affiliation:
The State University of Iowa, Iowa City, Iowa Cornell University, Ithaca, N.Y.
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Extract

In this paper a theorem will be established which states that a particular axiom of choice is equivalent to complete distributivity of union and intersection. The theorem will be formulated and proved in the system of logic of [4]. In addition to definitions of [4], the following will be used.

In terms of these definitions, the theorem can be formulated as follows.

The dual of this statement, obtained by interchanging I and U, is also a theorem and has a similar proof.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 4 , December 1954 , pp. 275 - 277
Copyright
Copyright © Association for Symbolic Logic 1954

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References

REFERENCES

[1]Birkhoff, Garrett, Lattice theory, American Mathematical Society Colloquium Publications, vol. 25, revised edition, 1948, pp. 146–147.Google Scholar
[2]Hahn, Hans, Reele Funktionen, Erster Teil, Leipzig, 1932, p. 10.Google Scholar
[3]Hausdorff, F., Mengenlehre, Dritte Auflage, 1935, p. 19.Google Scholar
[4]Quine, W. V., Mathematical logic, revised edition. Harvard, 1951.CrossRefGoogle Scholar
[5]Vaidyanathaswamy, R., Treatise on set topology, Part I, 1947, p. 3.Google Scholar