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Metamathematical problems

Published online by Cambridge University Press:  12 March 2014

Abraham Robinson
Abraham Robinson*
Affiliation:
Yale University, New Haven, Connecticut 06520 Institute for Advanced Study, Princeton, New Jersey 08540
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Extract

When a logician approaches the world of mathematics, he may have in mind one or more of several purposes. He may try to find in mathematics a framework for formalizing commonly accepted laws of thought or perhaps laws of thought that are not commonly accepted. He may want to assist the mathematician by providing him with firm foundations for his theories. But it may also be the case that the logician wishes to use his own characteristic tools—formalized languages, explicit relations between symbols and objects, rigidly expressed and controlled rules of deduction—in order to gain a better understanding of the various and variegated kinds of structures, methods, theories and theorems that are to be found in mathematics. We may then expect him to adopt the attitude of the physicist or psychologist who (whatever his professed philosophy) feels that he deals with phenomena of the external world, whose rules cannot be imposed by him arbitrarily. He, or those that come after him, may indeed use the understanding thus gained in order to modify these phenomena, but as a scientist he would not regard this possibility as his only justification.

For many years now, I have concentrated on the third of the lines of approach sketched above, and it seemed natural that I should discuss it again on the present occasion. However, today I do not wish to emphasize past developments but, using some of them as a background, I propose to enumerate a number of open problems. These problems seemed to me of some interest not only for their own sake but also because their solution might well require weapons whose introduction would close definite gaps in our armory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 3 , September 1973 , pp. 500 - 516
Copyright
Copyright © Association for Symbolic Logic 1973

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References

REFERENCES

[1]Arnol'd, V. I., Some questions of approximation and representation of functions, Proceedings of the International Congress of Mathematicians (Edinburgh, 1958), Cambridge University Press, New York, 1960, pp. 339–398. (Russian)Google Scholar
[2]Ax, J. and Kochen, S., Diophantine problems over local fields. I, II, American Journal of Mathematics, vol. 87 (1965), pp. 605–630, 631–648; Diophantine problems over local fields. III. Decidable fields, Annals of Mathematics, vol. 83 (1966), pp. 437–456.CrossRefGoogle Scholar
[3]Bernays, P., Zum Symposium über die Grundlagen der Mathematik, Dialectica, vol. 25 (1971), pp. 171–195.CrossRefGoogle Scholar
[4]Bowen, K. A., Forcing in a general setting, 1972 (unpublished).Google Scholar
[5]Brauer, R., A note on systems of homogeneous algebraic systems, Bulletin of the American Mathematical Society, vol. 51 (1945), pp. 749–755.CrossRefGoogle Scholar
[6]Cohen, P. J., Comments on the foundations of set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 9–15.Google Scholar
[7]Dedekind, R. and Weber, H., Theorie der algebraischen Functionen einer Veränderlichen, Journal für reine und angewandte Mathematik, vol. 92 (1882), pp. 181–290.Google Scholar
[8]Ershov, Y. L., On the elementary theory of maximal normal fields, Algebra i Logika, vol. 4 (1965), pp. 31–70.Google Scholar
[9]Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198.Google Scholar
[10]Gödel, K., Über die Länge von Beweisen, Ergebnisse eines mathematischen Kolloquiums, Fase. 7, 1936, pp. 23–24.Google Scholar
[11]Habicht, W., Über Polynomabbildungen, Mathematische Annalen, vol. 126 (1953), pp. 149–176.Google Scholar
[12]Herrmann, G., Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Mathematische Annalen, vol. 95 (1926), pp. 736–788.CrossRefGoogle Scholar
[13]Hilbert, D., Mathematische Probleme, Archiv für Mathematik und Physik (3), vol. 1 (1901), pp. 44–63, 213–237 (Collected works, vol. 3 (1930), pp. 290–329).Google Scholar
[14]Hille, È., Analytic function theory, Vol. II, Ginn, Boston, 1962, p. 250.Google Scholar
[15]Hirschfeld, J. M., Existentially complete and generic structures in arithmetic, Doctoral Dissertation, Yale University, New Haven, Conn., 1972.Google Scholar
[16]Johnson, R., The construction of elementary extensions and the Stone-Čech compactification, Doctoral Dissertation, Yale University, New Haven, Conn., 1971.Google Scholar
[17]Kolchin, E. R., Some problems in differential algebra, Proceedings of the International Congress of Mathematicians (Moscow 1966), “Mir”, Moscow, 1968, pp. 269–276.Google Scholar
[18]Kreisel, G., Mathematical significance of consistency proofs, this Journal, vol. 23 (1958), pp. 155–182.Google Scholar
[19]Kreisel, G., Observations on popular discussion of foundations, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 189–198.Google Scholar
[20]Lambert, W. M., A notion of effectiveness in arbitrary structures, this Journal, vol. 33 (1968), pp. 577–602.Google Scholar
[21]Laplace, P. S., Mémoire sur les solutions particulières des équations différentielles et sur les inégalités séculaires des planètes, (Collected works), vol. 8 (1772), pp. 326–365.Google Scholar
[22]Lightstone, A. H. and Robinson, Abraham, On the representation of Herbrand functions in algebraically closed fields, this Journal, vol. 22 (1957), pp. 187–204.Google Scholar
[23]Lorenzen, P., Constructive mathematics as a philosophical problem, Compositio Mathematica, vol. 20 (1968), pp. 133–142.Google Scholar
[24]Luxemburg, W. A. J. (Editor), Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston, New York, 1969.Google Scholar
[25]Macintyre, A., On algebraically closed groups, Annals of Mathematical Logic (to appear).Google Scholar
[26]McCord, M., Nonstandard analysis and homology, Fundamenta Mathematicae, vol. 74 (1972), pp. 21–28.CrossRefGoogle Scholar
[27]Morley, M., Applications of topology to Lω1ω, mimeographed, not dated.Google Scholar
[28]Post, E. L., Absolutely unsolvable problems and relatively undecidable propositions, 1941, The Undecidable (Davis, Martin, Editor), Raven Press, New York, 1965, pp. 338–433.Google Scholar
[29]Ritt, J. F., Differential algebra, American Mathematical Society Colloquium Publications, vol. 33, American Mathematical Society, Providence, R.I., 1950.Google Scholar
[30]Robinson, A., Proving a theorem (as done by man, logician, or machine), Summer Institute for Symbolic Logic (Cornell, 1957), Summaries, pp. 350–352.Google Scholar
[31]Robinson, A., Introduction to model theory and to the metamathematics of algebra, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1963.Google Scholar
[32]Robinson, A., Formalism '64, Proceedings of the International Congress for Logic, Methodology, and Philosophy of Science, North-Holland, Jerusalem 1964, pp. 228–246.Google Scholar
[33]Robinson, A., Nonstandard analysis, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam 1966.Google Scholar
[34]Robinson, A., Forcing in model theory, Symposia Mathematica, vol. 1, Academic Press, London, 1971, pp. 245–250.Google Scholar
[35]Robinson, A., Nonstandard arithmetic and generic arithmetic, Proceedings of the International Congress for Logic, Philosophy and Methodology of Science (Bucharest 1972) (to appear).Google Scholar
[36]Robinson, J., The decision problem for fields, Studies in Logic and the Foundations of Mathematics (Addison, J. W., Henkin, L., Tarski, A., Editors), North-Holland, Amsterdam 1965, pp. 299–311.Google Scholar
[37]Segre, B., Mathematical questions on algebraic varieties, The Athlone Press, London, 1957.Google Scholar
[38]Takeuti, G., Topological space and forcing, mimeographed, 1966.Google Scholar
[39]Tarski, A. and McKinsey, J. C. C., A decision method for elementary algebra and geometry, 1st edition, Rand Corp., Santa Monica, Calif., 1948; 2nd edition, University of California Press, Berkeley and Los Angeles, 1951.Google Scholar