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Zorn's lemma and complete Boolean algebras in intuitionistic type theories

Published online by Cambridge University Press:  12 March 2014

J. L. Bell*
Affiliation:
Department of Philosophy, The University of Western Ontario, London, Ontario, CanadaN6A 3K7, E-mail: jbell@julian.uwo.ca

Abstract

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some familiar results in the theory of (complete) Boolean algebras—notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Banaschewski, B., Compact regular frames and the Sikorski theorem, Kyungpook Mathematical Journal, vol. 28 (1988), no. 1, pp. 114.Google Scholar
[2]Banaschewski, B., On the injectivity of Boolean algebras, Commentationes Mathematiae Universitatis Carolinae, vol. 34 (1993), pp. 501511.Google Scholar
[3]Banaschewski, B. and Bhutani, K. R., Boolean algebras in a localic topos, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 100 (1986), pp. 4355.CrossRefGoogle Scholar
[4]Bell, J. L., On the strength of the Sikorski extension theorem for Boolean algebras, this Journal, vol. 48 (1983), no. 3, pp. 841846.Google Scholar
[5]Bell, J. L., Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Fundamenta Mathematicae, vol. 130 (1988), pp. 5155.CrossRefGoogle Scholar
[6]Bell, J. L., Toposes and local set theories: An introduction, Oxford University Press, 1988.Google Scholar
[7]Halmos, P. R., Lectures on Boolean algebras, Van Nostrand, 1963.Google Scholar
[8]Johnstone, P. T., Conditions related to de Morgan's law, Applications of sheaves, Springer Lecture Notes in Mathematics, no. 753, Springer-Verlag, 1979, pp. 479–91.CrossRefGoogle Scholar
[9]Monro, G. P., On generic extensions without the axiom of choice, this Journal, vol. 48 (1983), no. 1, pp. 3952.Google Scholar
[10]Sikorski, R., A theorem on extensions of homomorphisms, Annales de la Société Pol. de Mathématiques, vol. 21 (1948), pp. 332–35.Google Scholar