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MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS

Published online by Cambridge University Press:  12 July 2019

MARIANNE MORILLON
Affiliation:
LABORATOIRE D’INFORMATIQUE ET MATHÉMATIQUES PARC TECHNOLOGIQUE UNIVERSITAIRE BÂTIMENT 2, 2 RUE JOSEPH WETZELL, 97490 SAINTE CLOTILDE, FRANCE E-mail:marianne.morillon@univ-reunion.fr
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Abstract

In set theory without the Axiom of Choice, we consider Ingleton’s axiom which is the ultrametric counterpart of the Hahn–Banach axiom. We show that in ZFA, i.e., in the set theory without the Axiom of Choice weakened to allow “atoms,” Ingleton’s axiom does not imply the Axiom of Choice (this solves in ZFA a question raised by van Rooij, [27]). We also prove that in ZFA, the “multiple choice” axiom implies the Krein–Milman axiom. We deduce that, in ZFA, the conjunction of the Hahn–Banach, Ingleton and Krein–Milman axioms does not imply the Axiom of Choice.

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Copyright © The Association for Symbolic Logic 2019 

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References

Amice, Y., Les Nombres p-Adiques, Presses Universitaires de France, Paris, 1975. Collection SUP: Le Mathématicien, No. 14.Google Scholar
Bell, J. and Fremlin, D., A geometric form of the axiom of choice. Fundamenta Mathematicae, vol. 77 (1972), pp. 167170.CrossRefGoogle Scholar
Blass, A., Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society, vol. 255 (1979), pp. 3159.CrossRefGoogle Scholar
Dodu, J. and Morillon, M., The Hahn–Banach property and the Axiom of Choice. Mathematical Logic Quarterly, vol. 45 (1999), no. 3, pp. 299314.CrossRefGoogle Scholar
Fossy, J. and Morillon, M., The Baire category property and some notions of compactness. Journal of the London Mathematical Society II Series, vol. 57 (1998), no. 1, pp. 119.CrossRefGoogle Scholar
Goldblatt, R., On the role of the Baire category theorem and dependent choice in the foundations of logic, this Journal, vol. 50 (1985), pp. 412422.Google Scholar
Gouvêa, F. Q., p-Adic Numbers, Universitext. Springer-Verlag, Berlin, second edition, 1997.CrossRefGoogle Scholar
Halpern, J. D. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic Set Theory (Scott, D. S., editor), Proceedings of Symposia in Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1971, pp. 83134.CrossRefGoogle Scholar
Hodges, W., Läuchli’s algebraic closure of Q. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 79 (1976), no. 2, pp. 289297.CrossRefGoogle Scholar
Howard, P. E., Rado’s selection lemma does not imply the Boolean prime ideal theorem. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 30 (1984), no. 2, pp. 129132.CrossRefGoogle Scholar
Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
Ingleton, A. W., The Hahn-Banach theorem for nonArchimedean valued fields. Proceedings of the Cambridge Philosophical Society, vol. 48 (1952), pp. 4145.CrossRefGoogle Scholar
Jech, T. J., The Axiom of Choice, North-Holland, Amsterdam, 1973.Google Scholar
Kelley, J. L., The Tychonoff product theorem implies the axiom of choice. Fundamenta Mathematicae, vol. 37 (1950), pp. 7576.CrossRefGoogle Scholar
Kunen, K., Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
Läuchli, H., Auswahlaxiom in der Algebra. Commentarii Mathematici Helvetici, vol. 37 (1962/1963), pp. 118.CrossRefGoogle Scholar
Levy, A., Axioms of multiple choice. Fundamenta Mathematicae, vol. 50 (1962), pp. 475483.CrossRefGoogle Scholar
Luxemburg, W. A. J., Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem, Applications of Model Theory to Algebra, Analysis, and Probability (Luxemburg, W. A. J., editor), Holt, Rinehart and Winston, New York, 1969, pp. 123137.Google Scholar
Morillon, M., Some consequences of Rado’s selection lemma. Archive for Mathematical Logic, vol. 51 (2012), no. 7–8, pp. 739749.CrossRefGoogle Scholar
Morillon, M., Linear extenders and the axiom of choice.Commentationes Mathematicae Universitatis Carolinae, vol. 58 (2017), no. 4, pp. 419434.Google Scholar
Pawlikowski, J., The Hahn-Banach theorem implies the Banach-Tarski paradox. Fundamenta Mathematicae, vol. 138 (1991), no. 1, pp. 2122.CrossRefGoogle Scholar
Pincus, D., Independence of the prime ideal theorem from the Hahn Banach theorem. Bulletin of the American Mathematical Society, vol. 78 (1972), pp. 766770.CrossRefGoogle Scholar
Pincus, D., Adding dependent choice to the prime ideal theorem, Logic Colloquium 76 (Gandy, R. O. and Hyland, J. M. E., editors), Studies in Logic and Foundations of Mathematics, vol. 87, North-Holland, Amsterdam, 1977, pp. 547565.Google Scholar
Repický, M., A proof of the independence of the axiom of choice from the Boolean prime ideal theorem. Commentationes Mathematicae Universitatis Carolinae, vol. 56 (2015), no. 4, pp. 543546.CrossRefGoogle Scholar
Robert, A. M., A Course in p-Adic Analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000.CrossRefGoogle Scholar
van Rooij, A. C. M., NonArchimedean Functional Analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 51, Marcel Dekker, Inc., New York, 1978.Google Scholar
van Rooij, A. C. M., The axiom of choice in p-adic functional analysis, p-Adic Functional Analysis (Bayod, J. M., de Grande de Kimpe, N., and Martinez-Maurica, J., editors) Lecture Notes in Pure and Applied Mathematics, vol. 137, Dekker, New York, 1992, pp. 151156.Google Scholar

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