Geodesic
Tre percorsi nonstandard
Matematica, cultura e società, Série 1, Tome 8 (2023) no. 1, pp. 57-78.

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Introdotta da Abraham Robinson negli anni sessanta del secolo scorso, l'analisi nonstandard è oggi una disciplina che ha trovato una moltitudine di applicazioni in vari settori della matematica e oltre. In questo articolo, intendiamo fornire una panoramica di questa teoria trattandone tre aspetti diversi: uno didattico, uno fondazionale e uno applicato.
Introduced by Abraham Robinson in the 1960s, nonstandard analysis is today a discipline that has found many applications in various fields of mathematics and beyond. In this article, we intend to provide an overview of this theory by discussing three aspects: didactic, foundational, and applied. 
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Benci, Vieri; Luperi Baglini, Lorenzo. Tre percorsi nonstandard. Matematica, cultura e società, Série 1, Tome 8 (2023) no. 1, pp. 57-78. http://geodesic.mathdoc.fr/item/RUMI_2023_1_8_1_a4/

[1] S. Albeverio, J.E. Fenstad, R. Hoegh-Krohn, T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Dover Books on Mathematics,2009. | MR | Zbl

[2] A. Alexander, Infinitesimals: How a Dangerous Mathematical Theory Shaped the Modern World, Scientific American, 2014. | MR | Zbl

[3] V. Benci, M. Di Nasso, A ring homomorphism is enough to get nonstandard analysis, Bull. Belg. Math. Soc. 10 (2003), 1-10. | MR | Zbl

[4] V. Benci, M. Di Nasso, Alpha-theory: an elementary axiomatic for nonstandard analysis, Expo. Math. 21 (2003), 355-386. | DOI | MR | Zbl

[5] V. Benci, M. Di Nasso, Numerosities of labelled sets: a new way of counting, Adv. Math. 173 (2003), 50-67. | DOI | MR | Zbl

[6] V. Benci, M. Di Nasso, How to measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers, World Scientific, Singapore, 2018. | MR | Zbl

[7] V. Benci, P. Freguglia, La matematica e l'infinito, Carocci, 2019, ISBN: 8843095250.

[8] V. Benci, P. Freguglia, Alcune osservazioni sulla matematica non archimedea, Matem. Cultura e Soc., RUMI, 1 (2016), 105-122. | fulltext bdim | fulltext EuDML | Zbl

[9] V. Benci, L. Horsten, S. Wenmackers, Non-Archimedean Probability, Milan J. Math. 81 (2013), 121-151. | DOI | MR | Zbl

[10] V. Benci, L. Luperi Baglini, Euclidean numbers and numerosities, J. Symbolic Logic (2022), 1-35. | DOI | MR | Zbl

[11] V. Benci, Alla scoperta dei numeri infinitesimi, Lezioni di analisi matematica esposte in un campo non-archimedeo, Aracne editrice, Roma, 2018. | MR | Zbl

[12] V. Benci, Un possibile percorso didattico: dalle numerosità ai numeri euclidei, Atti del convegno: VIII giornata nazionale di Analisi Nonstandard (P. Bonavoglia ed.), 21-33, 2019, ISBN: 1220064475.

[13] P. Bontivoglia, Il calcolo infinitesimale. Analisi per i licei alla maniera non standard, Universal Book, 2011, ISBN9788896354131.

[14] G. Cherlin, J. Hirschfeld, Ultrafilters and ultraproducts in non-standard analysis, in: Contributions to NonStandard Analysis (W.A.J. Luxemburg, A. Robinson, eds.), North Holland, Amsterdam (1972), 261-279. | MR | Zbl

[15] M. Di Nasso, Iterated Hyper-Extensions and an Idempotent Ultrafilter Proof of Rado's Theorem, Proc. Amer.Math. Soc. 143 (2015), 1749-1761. | DOI | MR | Zbl

[16] M. Di Nasso, L. Luperi Baglini, Ramsey properties of nonlinear Diophantine equations, Adv. Math. 324 (2018),84-117. | DOI | MR

[17] M. Di Nasso, I. Goldbring, M. Lupini, Nonstandard Methods in Ramsey Theory and Combinatorial NumberTheory, Lecture Notes in Mathematics 2239, Springer, 2019. | DOI | MR | Zbl

[18] P. Du Bois-Reymond, Über die Paradoxen des Infinitär Calcüls, Math. Annalen 11 (1877), 150-167. | fulltext EuDML | DOI | MR | Zbl

[19] R. Goldblatt, Lectures on the Hyperreals – An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics, Vol. 188, Springer, New York, 1998. | DOI | MR | Zbl

[20] D. Hilbert, Grundlagen der Geometrie, Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen, Teubner, Leipzig, 1899. | MR | Zbl

[21] N. Hindman, D. Strauss, Algebra in the Stone-Čech Compactification, de Gruyter, Berlin, 1998. | DOI | MR | Zbl

[22] J. Hirschfeld, Nonstandard combinatorics, Studia Logica 47 (1988), 221-232. | DOI | MR

[23] K. Hrbáček, Internally iterated ultrapowers, in: Nonstandard Models of Arithmetic and Set Theory (A. Enayat, R. Kossak, eds.), Contemporary Math. 361, 2004, American Mathematical Society. | DOI | MR | Zbl

[24] K. Hrbáček, O. Lessmann, R. O'Donovan, Analysis with Ultrasmall Numbers, Chapman & Hall/CRC (2015). | Zbl

[25] R. Jin, Introduction of nonstandard methods for number theorists, Proceedings of the CANT (2005) – Conference in Honor of Mel Nathanson, Integers 8 (2008), A7. | MR | Zbl

[26] H.J. Keisler, Foundations of infinitesimal calculus, last edition, University of Wisconsin at Madison, 2009. | Zbl

[27] K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Log. 1 (1970), 179-227. | DOI | MR | Zbl

[28] S.C. Leth, Some Nonstandard Methods in Combinatorial Number Theory, Studia Logica 47 (1988), 265-278. | DOI | MR | Zbl

[29] T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, Venezia (Serie 7), (1892-93).

[30] L. Luperi Baglini, Hyperintegers and Nonstandard Techniques in Combinatorics of Numbers, PhD Dissertation (2012), University of Siena, arXiv: 1212.2049. | MR | Zbl

[31] L. Luperi Baglini, Quanti elementi ha un insieme infinito, Matematica Mente 293-94 (2022).

[32] L. Luperi Baglini, Nonstandard characterisations of tensor products and monads in the theory of ultrafilters, Math. Log. Quart. 65 (2019), 347-369. | DOI | MR | Zbl

[33] W.A.J. Luxemburg, A General Theory of Monads, in: Applications of Model Theory to Algebra, Analysis and Probability, (W.A.J. Luxemburg ed.), Holt, Rinehart, and Winston, New York, 1969, 18-86. | MR | Zbl

[34] E. Mendelson, Introduzione alla Logica Matematica, Bollati Boringhieri, 1977.

[35] G. Peano, Sugli ordini degli infiniti, Rendiconti della Reale Accademia dei Lincei 19 (1910), 778-781.

[36] A. Raab, An approach to nonstandard quantum mechanics, J. Math. Phys. 45 (2004), 4791-4809. | DOI | MR | Zbl

[37] A. Robinson, Non-standard Analysis, North Holland, Amsterdam, 1966. | MR | Zbl

[38] Y. Sun, Nonstandard analysis in mathematical economics, in: Nonstandard Analysis for the Working Mathematician (P. Loeb, M. Wolff eds.), Second edition, Springer (2015),349-399. | DOI | MR | Zbl

[39] G. Veronese, Il continuo rettilineo e l'assioma V di Archimede, Memorie della Reale Accademia dei Lincei, Atti della Classe di scienze naturali, fisiche e matematiche 4, (1889), 603-624. | Zbl

[40] G. Veronese, Intorno ad alcune osservazioni sui segmenti infiniti o infinitesimi attuali, Math. Ann. 47 (1896), 423-432. | fulltext EuDML | DOI | MR | Zbl

[41] D. Zambelli et al., Matematica C3, www.matematicadolce.eu, (2019). ISBN: 9788899988005. | Zbl