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Abstract
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The Notion of Set in Nonstandard Analysis
References

CREDENDA OF NONSTANDARD ANALYSIS

Profile image of Semen KutateladzeSemen Kutateladze

Abstract

The principal set-theoretic credos of nonstandard analysis are presented. A "naive" justification of the infinitesimal techniques and an overview of the corresponding formal apparatus are provided. The axioms of Nelson's internal set theory are discussed as well as those of the external set theories by Hrbaček and Kawai.

Key takeaways
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  1. The paper presents the foundational principles of nonstandard analysis, emphasizing the need for accessible methods.
  2. Two primary goals are outlined: simplifying nonstandard analysis concepts and providing axiomatic references.
  3. Nelson's internal set theory underpins nonstandard analysis, incorporating both standard and nonstandard sets.
  4. Key principles include the transfer principle and standardization, which aid in linking standard and nonstandard objects.
  5. The historical evolution of mathematical analysis and its founders is discussed, highlighting contributions from Leibniz, Newton, and others.

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References (81)

  1. ALEXANDROV A. D., A General View of Mathematics, in Mathematics, Its Contents, Meth- ods and Significance, 1956, pp. 5-78.
  2. ARCHIMEDES, Collected Works, Moscow, 1962.
  3. BERKELEY G., Collected Works, Mysl', Moscow, 1978.
  4. BLEKHMAN I. I., MYSHKIS A. D., and PANOVKO A. G., Mechanics and Applied Mathe- matics. Logic and the Specifics of Mathematical Applications, Nauka, Moscow, 1983.
  5. BOGOLYUBOV A. N., Read Euler: He is Our Common Teacher, Science in the USSR 6 (1984), 98-104.
  6. BOREL E., Probability and Certainty, GIFML, Moscow, 1961.
  7. VILENKIN I., The Commodore of "Luzitania", Znanie-Sila (1984), no. 1, 27-29.
  8. VOLTAIRE F.-M., Micromegas 1752, in Philosophical Novels, GIHL, Moscow, 1960.
  9. VOPENKA P., Mathematics in the Alternative Set Theory, Mir Publishers, Moscow.
  10. GOLDBLATT R., Topoi. The Categorial Analysis of Logic, Mir, Moscow.
  11. DAVIS M., Applied Nonstandard Analysis, Mir, Moscow, 1980.
  12. YESENIN-VOLPIN A. S., An analysis of potential realizability, Studies in Logic, AN SSSR, Moscow, 1959, pp. 218-262.
  13. ZORICH V. A., Mathematical Analysis, vol. 1, Nauka, Moscow, 1981.
  14. ZVONKIN A. K. and SHUBIN M. A., Nonstandard analysis and singular perturbations of ordinary differential equations, Uspekhi Mat. Nauk 39 (1984), no. 2, 77-127.
  15. CANTOR G., Papers on Set Theory, Nauka, Moscow, 1985.
  16. KOLMOGOROV A. N. and DRAGALIN A. G., Mathematical Logic. Supplementary Chap- ters, Moscow University, 1984.
  17. COHEN P. J., On Foundation of Set Theory, Uspekhi Mat. Nauk 29 (1972), no. 5, 169-176.
  18. COHEN P. J., Set Theory and the Continuum Hypothesis, Mir, Moscow, 1969.
  19. KUTATELADZE S. S., Foundations of Nonstandard Analysis, Novosibirsk University, 1986.
  20. COURANT R., Differential and Integral Calculus, vol. 1, Nauka, Moscow, 1967.
  21. KURATOWSKI K. and MOSTOWSKI A., Set Theory, Mir, Moscow, 1970.
  22. LAVRENT'EV M. A., Science. Technology. Personnel, Nauka, Novosibirsk, 1980.
  23. LAVRENT'EV M. A., NIKOLAI NIKOLAIEVICH LUZIN, Uspekhi Mat. Nauk 29 (1979), no. 5, 177-182.
  24. LEGE J.-M., Science, Technology and the World, Nauka i Zhizn' (1986), no. 11, 3-11.
  25. LEIBNIZ G. W., Nova Methodus pro Maximis et Minimis, Itemque Tangentibus, quae nec Fractas nec Irrationales Quantitates Moratur, et Singulare pro Illis Calculi Genus, Uspekhi Mat. Nauk 3 (1948), no. 1, 166-173.
  26. LENIN V. I., Collected Works, vol. 18, GIPL, Moscow.
  27. LENIN V. I., Collected Works, vol. 29, GIPL, Moscow.
  28. N. N. Luzin-an outstanding mathematician and teacher, Vestnik AN SSSR (1984), no. 11, 95-102.
  29. LUZIN N. N., Differential Calculus, Vyshaya Shkola, Moscow, 1961.
  30. LUZIN N. N., Collected Works, vol. 3, Izd. AN SSSR, Moscow, 1959.
  31. LUZIN N. N., Theory of Functions of a Real Variable, GUPI, Moscow, 1940.
  32. LYANTZE V. E., On nonstandard analysis, in Mathematics today, Vyshcha Shkola, Kiev, 1986, pp. 26-44.
  33. MAL TSEV A. I., Untersuchungen aus dem Gebiete der matematischen Logik, Mat. Sb. 1 (1976), no. 3, 323-336.
  34. MANIN YU. I., Provable and Unprovable, Sovetskoe Radio, Moscow, 1979.
  35. MARX K., Mathematical Manuscripts, Nauka, Moscow, 1968.
  36. MOLCHANOV V. A., Introduction to the Calculus of Infinitesimals, Saratov Ped. Institute, 1986.
  37. EUCLID, Fundamentals, vol. 8-10, GITTL, Moscow-Leningrad, 1949.
  38. NEWTON I., Mathematical Papers, ONTI, Moscow-Leningrad, 1937.
  39. ROBINSON A., Introduction to Model Theory and to the Metamathematics of Algebra, Nauka, Moscow, 1967.
  40. RUZAVIN G. I., Philosophical Problems in the Foundations of Mathematics, Nauka, Moscow, 1983.
  41. STROYAN K. D., Infinitesimal analysis of curves and surfaces, in: The Reference Book on Mathematical Logic, vol. 1, Nauka, Moscow, 1982, pp. 199-234.
  42. USPENSKII V. A., What Is Nonstandard Analysis, Nauka, Moscow, 1987.
  43. FRAENKEL A. and BAR-HILLEL I., Foundations of Set Theory, Mir, Moscow, 1966.
  44. A Textbook on the History of Mathematics, Prosweshchenie, Moscow, 1977.
  45. SCHOENFILD J. R., Axioms of set theory, in The Reference Book on Mathematical Logic, vol. 2, Nauka, Moscow, 1982, pp. 9-34.
  46. EULER L., Introduction to Infinitesimal Analysis, vol. 1, ONTI, Moscow, 1936.
  47. EULER L., Differential Calculus, ONTI, Moscow, 1949.
  48. EULER L., Integral Calculus, ONTI, Moscow, 1949.
  49. ECKLOF P., Theory of ultraproducts for algebraists, in: The Reference Book on Mathematical Logic, vol. 1, Nauka, Moscow, 1982, pp. 109-140.
  50. ENGELS F., The Dialectics of Nature, in: Marx K. and Engels F. Collected works, vol. 20, GIPL, Moscow, 1961, pp. 339-626.
  51. YUSHKEWICH A. P., Leibniz and the foundation of infinitesimal calculus, Uspekhi Mat. Nauk 3 (1948), no. 1, 150-164.
  52. ALBERIO S., FENSTAD J. F. et al., Nonstandard Methods in Stochastic Analysis and Math- ematical Physics, Academic Press, Orlando, 1986.
  53. Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rhinehart and Winston, New York, 1966.
  54. BELL J. L. and SLOMSON A. B., Models and Ultraproducts. An Introduction, North-Holland, Amsterdam-London, 1971.
  55. Contributions to Nonstandard Analysis, North-Holland, Amsterdam, 1972.
  56. CUTLAND N., Nonstandard Measure theory and its applications, Bull. London Math. Soc. 15 (1983), no. 6, 530-589.
  57. GANDY R. O., Limitations to mathematical knowledge, in: Logic Colloquiuum-80, North- Holland, New York-London, 1982, pp. 129-146.
  58. G ÖDEL K, What is Cantor's continuum problem, Amer. Math. Monthly 54 (1947), no. 9, 515-525.
  59. HARNIK V., Infinitesimals from Leibniz to Robinson time-to bring them back to school, Math. Intelligencer 8 (1986), no. 2, 41-47.
  60. HENLE J. M. and KLEINBERG E. M., Infinitesimal Calculus, Alpine Press, Cambridge- London, 1979.
  61. HENSON W. and KEISLER J., On the strength of nonstandard analysis, J. Symb. Logic 51 (1986), no. 2, 377-386.
  62. HRBA ČEK K., Axiomatic foundations for nonstandard analysis, Fund. Math. 98 (1978), no. 1, 1-24.
  63. HRBA ČEK K., Nonstandard set theory, Amer. Math. Monthly 86 (1979), no. 8, 659-677.
  64. HURD A. E. and LOEB H., Introduction to Nonstandard Analysis, Academic Press, Orlando, 1985.
  65. JARNIK V., Bolzano and the Foundations of MAthematical Analysis, Society of Czechoslovak. Math. Phys., Prague, 1981.
  66. JAFFE A., Ordering the universe: the role of mathematics, SIAM Review 26 (1984), no. 4, 473-500.
  67. KAWAI T., Axiom system of nonstandard set theory, in Logic Symposia, Hakone 1979, 1980, Springer, Berlin, 1981, pp. 57-65.
  68. KEISLER J., An Infinitesimal Approach to Stochastic Analysis, vol. 48, Memoir. Amer. Math. Soc., 1984.
  69. KOPPERMAN R., Model Theory and its Applications, Allyn and Bacon, Boston, 1972.
  70. KREISEL G., Observations of popular discussions on foundations, in Axiomatic Set Theory. Proc. of Symposia in Pure Math., vol. 1, Amer. Math. Soc., Providence, 1971, pp. 183-190.
  71. LEVY A, Basic Set Theory, Springer, Berlin, 1979.
  72. LUTZ R. and GOZE M., Nonstandard Analysis. A Practical Guide with Application Lecture Notes in Math. 881, Springer, Berlin, 1981.
  73. MOCHOVER M. and HIRSCHFELD J., Lectures on Nonstandard Analysis Lecture Notes in Math. 94, Springer, Berlin, 1969.
  74. NELSON E., Internal set theory. A new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165-1198.
  75. Nonstandard Analysis. Recent Developments, Springer, Berlin, 1983 Lecture Notes in Math. 983.
  76. ROBINSON A., Nonstandard Analysis, North-Holland, Amsterdam-London,, 1970.
  77. ROBINSON A., The metaphysics of the calculus, in Problems in the Philosophy of Mathe- matics, vol. 1, North-Holland, Amsterdam, 1967, pp. 28-46.
  78. STROYAN K. D. and BAYOD J. M., Foundations of Infinitesimal Stochastic Analysis, North- Holland, Amsterdam, 1986.
  79. STROYAN K. D. and LUXEMBERG W. A. J., Introduction to the Theory of Infinitesimals, Academic Press, N. Y., 1976.
  80. WESTFALL R., Never at Rest. A Bibliography of Isaac Newton, Cambridge Univ. Press, Cambridge, 1982.
  81. ZIVALJEVIC R., Infinitesimals, microsimplexes and elementary homology theory, Amer. Math. Monthly 93 (1986), no. 7, 540-544.

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