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MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS

  • MARIANNE MORILLON (a1)

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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[1]Amice, Y., Les Nombres p-Adiques, Presses Universitaires de France, Paris, 1975. Collection SUP: Le Mathématicien, No. 14.
[2]Bell, J. and Fremlin, D., A geometric form of the axiom of choice. Fundamenta Mathematicae, vol. 77 (1972), pp. 167170.
[3]Blass, A., Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society, vol. 255 (1979), pp. 3159.
[4]Dodu, J. and Morillon, M., The Hahn–Banach property and the Axiom of Choice. Mathematical Logic Quarterly, vol. 45 (1999), no. 3, pp. 299314.
[5]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.
[6]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.
[7]Gouvêa, F. Q., p-Adic Numbers, Universitext. Springer-Verlag, Berlin, second edition, 1997.
[8]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.
[9]Hodges, W., Läuchli’s algebraic closure of Q. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 79 (1976), no. 2, pp. 289297.
[10]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.
[11]Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998.
[12]Ingleton, A. W., The Hahn-Banach theorem for nonArchimedean valued fields. Proceedings of the Cambridge Philosophical Society, vol. 48 (1952), pp. 4145.
[13]Jech, T. J., The Axiom of Choice, North-Holland, Amsterdam, 1973.
[14]Kelley, J. L., The Tychonoff product theorem implies the axiom of choice. Fundamenta Mathematicae, vol. 37 (1950), pp. 7576.
[15]Kunen, K., Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.
[16]Läuchli, H., Auswahlaxiom in der Algebra. Commentarii Mathematici Helvetici, vol. 37 (1962/1963), pp. 118.
[17]Levy, A., Axioms of multiple choice. Fundamenta Mathematicae, vol. 50 (1962), pp. 475483.
[18]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.
[19]Morillon, M., Some consequences of Rado’s selection lemma. Archive for Mathematical Logic, vol. 51 (2012), no. 7–8, pp. 739749.
[20]Morillon, M., Linear extenders and the axiom of choice.Commentationes Mathematicae Universitatis Carolinae, vol. 58 (2017), no. 4, pp. 419434.
[21]Pawlikowski, J., The Hahn-Banach theorem implies the Banach-Tarski paradox. Fundamenta Mathematicae, vol. 138 (1991), no. 1, pp. 2122.
[22]Pincus, D., Independence of the prime ideal theorem from the Hahn Banach theorem. Bulletin of the American Mathematical Society, vol. 78 (1972), pp. 766770.
[23]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.
[24]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.
[25]Robert, A. M., A Course in p-Adic Analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000.
[26]van Rooij, A. C. M., NonArchimedean Functional Analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 51, Marcel Dekker, Inc., New York, 1978.
[27]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.

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MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS

  • MARIANNE MORILLON (a1)

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