Hostname: page-component-89b8bd64d-b5k59 Total loading time: 0 Render date: 2026-05-10T13:56:41.850Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS

Published online by Cambridge University Press:  12 July 2019

MARIANNE MORILLON
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
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Primary 03E25Secondary 46S10Axiom of Choicenonarchimedean fieldsIngleton’s theoremKrein–Milman’s theorem

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 439 - 455
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

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. 167–170.CrossRefGoogle Scholar
Blass, A., Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society, vol. 255 (1979), pp. 31–59.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. 299–314.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. 1–19.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. 412–422.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. 83–134.CrossRefGoogle Scholar
Hodges, W., Läuchli’s algebraic closure of Q. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 79 (1976), no. 2, pp. 289–297.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. 129–132.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. 41–45.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. 75–76.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. 1–18.CrossRefGoogle Scholar
Levy, A., Axioms of multiple choice. Fundamenta Mathematicae, vol. 50 (1962), pp. 475–483.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. 123–137.Google Scholar
Morillon, M., Some consequences of Rado’s selection lemma. Archive for Mathematical Logic, vol. 51 (2012), no. 7–8, pp. 739–749.CrossRefGoogle Scholar
Morillon, M., Linear extenders and the axiom of choice.Commentationes Mathematicae Universitatis Carolinae, vol. 58 (2017), no. 4, pp. 419–434.Google Scholar
Pawlikowski, J., The Hahn-Banach theorem implies the Banach-Tarski paradox. Fundamenta Mathematicae, vol. 138 (1991), no. 1, pp. 21–22.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. 766–770.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. 547–565.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. 543–546.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. 151–156.Google Scholar