Abstract
It is now over forty years since Abraham Robinson realized that “the concepts and methods of Mathematical Logic are capable of providing a suitable framework fur the development of the Differential and Integral Calculus by means of infinitely small and infinitely large numbers” (Robinson [29], Introduction, p. 2). The magnitude of Robinson’s achievement cannot be overstated. Not only does his framework allow rigorous paraphrases of many arguments of Leibniz, Euler and other mathematicians from the classical period of calculus; it has enabled the development of entirely new, important mathematical techniques and constructs not anticipated by the classics. Researchers working with the methods of nonstandard analysis have discovered new significant results in diverse areas of pure and applied mathematics, from number theory to mathematical physics and economics.
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References
P.V. Andreev, The notion of relative standardness in axiomatic systems of nonstandard analysis, Thesis, Dept. Math. Mech., State University of N.I. Lobachcvskij in Nizhnij Novgorod, 2002, 96 pp. (Russian).
P.V. Andreev, “On definable standardness predicates in internal set theory”, Mathematical Notes, 66 (1999) 803–809 (Russian).
D. Ballard, Foundational Aspects of “Non” standard Mathematics, Contemporary Mathematics, vol. 176, American Mathematical Society, Providence, R.I., 1994.
V. Benci and M. Di Nasso, “Alpha-theory: an elementary axiomatics for nonstandard analysis”, Expositiones Math., 21 (2003) 355–386.
B. Benninghofen and M.M. Richter, “A general theory of superin-finitesimals”, Fund. Math., 123 (1987) 199–215.
C.C. Chang and H.J. Keisler, Model Theory, 3rd edition, North-Holland Publ. Co., 1990.
G. Cherlin and J. Hirschfeld, Ultrafilters and ultraproducts in non-standard analysis, in Contributions to Non-standard Analysis, ed. by W.A.J. Luxemburg and A. Robinson, North Holland, Amsterdam 1972.
N.J. Cutland, “Transfer theorems for π-monads”, Ann. Pure Appl. Logic, 44 (1989) 53–62.
P. Fletcher, “Nonstandard set theory”, J. Symbolic Logic, 54 (1989) 1000–1008.
R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Graduate Texts in Math. 188, Springer-Verlag, New York, 1998.
E.I. Gordon, “Relatively nonstandard elements in the theory of internal sets of E. Nelson”, Siberian Mathematical Journal, 30 (1989) 89–95 (Russian).
E.I. Gordon, Nonstandard Methods in Commutative Harmonic Analysis, American Mathematical Society, Providence, Rhode Island, 1997.
K. Hrbacek, “Axiomatic foundations for nonstandard analysis”, Funda-menta Mathematicae, 98 (1978) 1–19; abstract in J. Symbolic Logic, 41 (1976) 285.
K. Hrbacek, “Nonstandard set theory”, Amer. Math. Monthly, 86 (1979) 1–19.
K. Hrbacek, Internally iterated ultrapowers, in Nonstandard Models of Arithmetic and Set Theory, ed. by A. Enayat and R. Kossak, Contemporary Math. 361, American Mathematical Society, Providence, R.I., 2004.
K. Hrbacek, Nonstandard objects in set theory, in Nonstandard Methods and Applications in Mathematics, ed. by N.J. Cutland, M. Di Nasso and D.A. Ross, Lecture Notes in Logic 25, Association for Symbolic Logic, Pasadena, CA., 2005, 41 pp.
K. Hrbacek, Relative Set Theory, work in progress.
V. Kanovei and M. Reeken, “Internal approach to external sets and universes”, Studia Logica, Part I, 55 (1995) 227–235; Part II, 55 (1995) 347–376; Part III, 56 (1996) 293–322.
V. Kanovei and M. Reeken, Nonstandard Analysis, Axiomatically, Springer-Verlag, Berlin, Heidelberg, New York, 2004.
H. J. Keisler, Calculus: An Infinitesimal Approach, Prindle, Weber and Scmidt, 1976, 1986.
W.A.J. Luxemburg, A general theory of monads, in: W.A.J. Luxemburg, ed., Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston 1969.
V.A. Molchanov, “On applications of double nonstandard enlargements to topology”, Sibirsk. Mat. Zh., 30 (1989) 64–71.
E. Nelson, “Internal set theory: a new approach to Nonstandard Analysis”, Bull. Amer. Math. Soc., 83 (1977) 1165–1198.
Y. Péraire and G. Wallet, “Une théorie relative des ensembles intérnes”, C.R. Acad. Sci. Paris, Sér. I, 308 (1989) 301–304.
Y. Péraire, “Théorie relative des ensembles intérnes”, Osaka Journ. Math., 29 (1992) 267–297.
Y. Péraire, “Some extensions of the principles of idealization transfer and choice in the relative internal set theory”, Arch. Math. Logic, 34 (1995) 269–277.
Y. Péraire, “Formules absolues dans la théorie relative des ensembles intérnes”, Rivista di Matematica Pura ed Applicata, 19 (1996) 27–56.
Y. Péraire, “Infinitesimal approach of almost-automorphic functions”, Annals of Pure and Applied Logic, 63 (1993) 283–297.
A. Robinson, Non-standard Analysis, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1966.
A. Robinson and E. Zakon, A set-theoretical characterization of enlargements, in W.A.J. Luxemburg, ed., Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston 1969.
K.D. Stroyan, Foundations of Infinitesimal Calculus, 2nd ed., Academic Press 1997.
K.D. Stroyan, B. Benninghofen and M.M. Richter, “Superinfinitesimals in topology and functional analysis”, Proc. London Math. Soc., 59 (1989) 153–181.
K.D. Stroyan, Superinfinitesimals and inductive limits, in Nonstandard Analysis and its Applications, ed. by N. Cutland. Cambridge University Press, New York, 1988.
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Hrbacek, K. (2007). Stratified analysis?. In: van den Berg, I., Neves, V. (eds) The Strength of Nonstandard Analysis. Springer, Vienna. https://doi.org/10.1007/978-3-211-49905-4_4
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