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APPROXIMATING BEPPO LEVI’S PRINCIPIO DI APPROSSIMAZIONE

Published online by Cambridge University Press:  26 June 2014

RICCARDO BRUNI  and
PETER SCHUSTER
RICCARDO BRUNI
Affiliation:
DEPT. OF PHILOSOPHY, UNIVERSITY OF FLORENCE FIRENZE 50134, ITALYE-Mail:[email protected]
PETER SCHUSTER
Affiliation:
DEPT. OF PURE MATHEMATICS, UNIVERSITY OF LEEDS LEEDS LS2 9JT, UKE-Mail:[email protected]
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Abstract

We try to recast in modern terms a choice principle conceived by Beppo Levi, who called it the Approximation Principle (AP). Up to now, there was almost no discussion about Levi’s contribution, due to the quite obscure formulation of AP the author has chosen. After briefly reviewing the historical and philosophical surroundings of Levi’s proposal, we undertake our own attempt at interpreting AP. The idea underlying the principle, as well as the supposed faithfulness of our version to Levi’s original intention, are then discussed. Finally, an application of AP to a property of metric spaces is presented, with the aim of showing how AP may work in contexts where other forms of choice are commonly at use.

Keywords

Principle of approximationaxiom of choicedeductive domainscountably coverable open subsets (of a metric space)
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 2 , June 2014 , pp. 141 - 169
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Beeson, M.J., Foundations of Constructive Mathematics, Springer, Berlin, 1985.CrossRefGoogle Scholar
Bernstein, F., Untersuchungen aus der Mengenlehre. Halle a.d.S., Göttingen, 1901. Printed in Mathematische Annalen, vol. 61 (1905), pp. 117–155.CrossRefGoogle Scholar
Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.Google Scholar
Bishop, E. and Bridges, D., Constructive Analysis, Springer, Berlin, 1967.Google Scholar
Borel, E., Leçons sur le fonctions monogénes uniforme d’une variable complexe, Gauthier–Villars, Paris, 1917.Google Scholar
Bruni, R., Beppo Levi’s analysis of the paradoxes. Logica Universalis, vol. 7 (2013), pp. 211–231.CrossRefGoogle Scholar
Carathéodory, C., Vorlesungen über reelle Funktionen, Teubner, Leipzig u. Berlin, 1918.Google Scholar
Cassina, U., Sul principio della scelta ed alcuni problemi dell’infinito. Rendiconti del Seminario Matematico e Fisico di Milano, vol. 10 (1936), pp. 53–81.CrossRefGoogle Scholar
Faedo, A., Il principio di Zermelo per gli spazi astratti. Annali della Scuola Normale Superiore di Pisa, II serie, vol. 9 (1940), pp. 263–276.Google Scholar
Fitting, M., First-order Logic and Automated Theorem Proving, second edition, Springer, New York, 1996.CrossRefGoogle Scholar
Herrlich, H., Axiom of Choice, Springer, Berlin, 2006.Google Scholar
Hilbert, D., Die logischen Grundlagen der Mathematik. Mathematische Annalen, vol. 88 (1922), pp. 151–165.CrossRefGoogle Scholar
Hilbert, D., Neubegründung der Mathematik. Abhandlungen aus dem Mathematischen Seminar der Hamburger Universität, vol. 1 (1922), pp. 157–177.CrossRefGoogle Scholar
Hilbert, D., Ricerche sui Fondamenti della Matematica (Abrusci, V.M., editor), Bibliopolis, Napoli, 1985.Google Scholar
Howard, P. and Rubin, J.E., Consequences of the Axiom of Choice, American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
Jech, T., The Axiom of Choice, North-Holland Publishing Co., Amsterdam, 1973.Google Scholar
Jech, T., Set Theory, Springer, Berlin, 1997.CrossRefGoogle Scholar
Levi, B., Intorno alla teoria degli aggregati. Rendiconti del R. Ist. Lomb. di Sc. e Lett., serie II, vol. 35 (1902), pp. 863–868. Reprinted in 25, pp. 177–182.Google Scholar
Levi, B., Antinomie logiche? Annali di Matematica Pura ed Applicata, (III), vol. 15 (1908),pp. 187–216. Reprinted in 25, pp. 629–658.CrossRefGoogle Scholar
Levi, B., Introduzione all’Analisi Matematica, A. Herman et fils, Paris, 1916.Google Scholar
Levi, B., Riflessioni sopra alcuni principii della teoria degli aggregati e delle funzioni. In Scritti Matematici offerti a Enrico d’Ovidio, pp. 305–324. Torino, Bocca, 1918. Reprinted in 25, pp. 791–810.Google Scholar
Levi, B., Sui procedimenti transfiniti (Auszug aus einem Briefe an Herrn Hilbert). Mathematische Annalen, vol. 90 (1923), pp. 164–173. Reprinted in 25, pp. 869–878.Google Scholar
Levi, B., La nozione di “dominio deduttivo” e la sua importanza in taluni argomenti relativi ai fondamenti dell’analisi. Fundamenta Mathematicae, vol. 23 (1934), pp. 63–74.CrossRefGoogle Scholar
Levi, B., La noción de “dominio deductivo” como elemento de orientación en las cuestiones de fundamentos de las teorias matemáticas. Publicaciones del Instituto de Matemática (Rosario, AR), vol. 2 (1940), pp. 177–208.Google Scholar
Levi, B., Opere 1897–1926 (Lolli, G. et al., editors), vol. 1–2, Cremonese, Roma, 1999.Google Scholar
Levi, B. and Viola, T., Intorno ad un ragionamento fondamentale nella teoria delle famiglie normali di funzioni. Bollettino della Unione Matematica Italiana, vol. 12 (1933), no. 4, pp. 197–203.Google Scholar
Lévy, A., Basic Set Theory, Springer, Berlin, 1979.CrossRefGoogle Scholar
Lolli, G., A Berry–type paradox. In Randomness and Complexity. From Leibniz to Chaitin (Calude, Cristian S., editor), World Scientific Publishing Co., Singapore, 2007, pp. 155–159.CrossRefGoogle Scholar
Lubarski, R., Richman, F., and Schuster, P., The Kripke Schema in metric topology. Mathematical Logic Quarterly, vol. 58 (2012), pp. 498–501.CrossRefGoogle Scholar
Moore, G.H., Zermelo’s Axiom of Choice: its Origin, Development, and Influence, Springer, Berlin, 1982.CrossRefGoogle Scholar
Peano, G., Démonstration de l’intégrabilité des équations différentielles ordinaires. Mathematische Annalen, vol. 37 (1890), no. 2, pp. 182–228.CrossRefGoogle Scholar
Quine, W.V., Review of: La noción de ‘dominio deductivo’ como elemento de orientación en las cuestiones de fundamentos de las teorias matemáticas. by Beppo Levi. The Journal of Symbolic Logic, vol. 7 (1942), pp. 44–45.CrossRefGoogle Scholar
Rathjen, M., The disjunction and related properties for constructive Zermelo–Fraenkel set theory. The Journal of Symbolic Logic, vol. 70 (2005), pp. 1233–1254.CrossRefGoogle Scholar
Rathjen, M., From the weak to the strong existence property. Annals of Pure and Applied Logic, vol. 163 (2012), pp. 1400–1418.CrossRefGoogle Scholar
Reid, C., Hilbert, Springer, Berlin, 1970.CrossRefGoogle Scholar
Rubin, H. and Rubin, J.E., Equivalents of the Axiom of Choice, North-Holland Publishing Co., Amsterdam, 1963.Google Scholar
Rubin, H. and Rubin, J.E., Equivalents of the Axiom of Choice, North-Holland Publishing Co., Amsterdam, 1970. 2nd edition of 36.Google Scholar
Rubin, H. and Rubin, J.E., Equivalents of the Axiom of Choice. II, North-Holland Publishing Co., Amsterdam, 1985.Google Scholar
Schuster, P., Countable choice as a questionable uniformity principle. Philosophia Mathematicae, vol. 12 (2004), pp. 106–134 .CrossRefGoogle Scholar
Schuster, P. and Zappe, J., Über das Kripke-Schema und abzählbare Teilmengen. Logique et Analyse (N.S.), vol. 51 (2008), pp. 317–329.Google Scholar
Scorza-Dragoni, G., Sull’approssimazione dell’integrale di Lebesgue mediante integrale di Riemann. Annali di Matematica Pura ed Applicata, vol. 7 (1930), no. 4, pp. 61–70.CrossRefGoogle Scholar
Scorza-Dragoni, G., Sul principio di approssimazione nella teoria degli insiemi e sulla quasi–continuità delle funzioni misurabili. Rendiconti del Seminario Matematico della Regia Università di Roma, vol. 1 (1936), pp. 53–58.Google Scholar
Simpson, S.G., Subsystems of Second–Order Arithmetic, Springer, Berlin, 1999.CrossRefGoogle Scholar
Viola, T., Riflessioni intorno ad alcune applicazioni del postulato della scelta di E. Zermelo e del principio di approssimazione di B. Levi nella teoria degli aggregati. Bollettino della Unione Matematica Italiana, vol. 10 (1931), no. 5, pp. 287–294.Google Scholar
Viola, T., Sul principio di approssimazione di B. Levi nella teoria della misura e degli aggregati e in quella dell’integrale di Lebesgue. Bollettino della Unione Matematica Italiana, vol. 11 (1932), no. 2, pp. 74–78.Google Scholar
Viola, T., Ricerche assiomatiche sulle teorie delle funzioni d’insieme e dell’integrale di Lebesgue. Fundamenta Mathematicae, vol. 23 (1934), pp. 74–101.CrossRefGoogle Scholar
Weihrauch, K., Computable Analysis, Springer, Berlin, 2000.CrossRefGoogle Scholar
Zermelo, E., Beweis daß jede Menge wohlgeordnet werden kann. Mathematische Annalen, vol. 59 (1904), pp. 514–516.CrossRefGoogle Scholar
Zermelo, E., Neuer Beweis für die Möglichkeit einer Wohlordnung. Mathematische Annalen, vol. 65 (1908), pp. 107–128.CrossRefGoogle Scholar