# Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics

Chapter
Part of the Synthese Library book series (SYLI, volume 251)

## Abstract

In this paper, I will discuss a conceptual distinction, or a contrast, between two opposing ideas, that has played an extremely important role in the foundations of mathematics. The distinction was first formulated explicitly, though not quite generally, by Leon Henkin in 1950.1 He called it a distinction between the standard and the nonstandard interpretation of higher-order logic. I will follow his terminology, even though it may not be the most fortunate one. One reason for saying that this nomenclature is not entirely happy is that it is not clear which interpretation is the “standard” one in the sense of being a more common one historically. Another reason is that one can easily characterize more than one nonstandard interpretation of higher-order logic, even though Henkin considered only one.2 Furthermore, it turns out that the distinction (rightly understood) is not restricted to higher-order logics.’ Last but not least, it is far from clear how Henkin’s notion of standard interpretation or standard model is related to logicians’ idea of the standard model of such first-order theories as elementary arithmetic, in which usage “standard model” means simply “intended model”.

## Keywords

Arbitrary Function Standard Interpretation Propositional Function Intended Model Nonstandard Interpretation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## NOTES

1. 1.
Leon Henkin: 1950, Completeness in the Theory of Types, Journal of Symbolic Logic 15 81–91.Google Scholar
2. For a correction to Henkin’s paper, see Peter B. Andrews: 1972, General Models and Extensionality, Journal of Symbolic Logic 37, 395–397.Google Scholar
3. 2.
4. 3.
Cf. here Jaakko Hintikka: 1980, Standard vs. Nonstandard Logic: Higher Order, Modal and First-Order Logics, in E. Agazzi (ed.), Modern Logic: A Survey, D. Reidel, Dordrecht, 283–296.Google Scholar
5. 4.
The following paragraphs as well as section 3 below follow closely the exposition in Jaakko Hintikka and Gabriel Sandu: 1992, The Skeleton in Frege’s Cupboard: The Standard versus Nonstandard Distinction, Journal of Philosophy 89 290–315.Google Scholar
6. 5.
See Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, in Solomon Feferman et al. (eds.), Kurt Gödel: Collected Works, Vol. 2: Publications 1938–1974, Oxford University Press, New York, 1990, pp. 240–251. (Cf. also pp. 217–241.)Google Scholar
7. 6.
Cf., e.g., Jaakko Hintikka: 1955, `Reductions in the Theory of Types’, Acta Philosophica Fennica 559–115.Google Scholar
8. 7.
Cf. here Hintikka and Sandu, op. cit. note 4.Google Scholar
9. 8.
Cf. the end of section 4 below.Google Scholar
10. 9.
This question is tantamount to the question of the validity of Leibniz’s Law.Google Scholar
11. 10.
Quoted in Gregory H. Moore: 1982, Zermelo’s Axiom of Choice, Springer-Verlag, Berlin-Heidelberg-New York, p. 314.Google Scholar
12. 11.
Quoted in op. cit., p. 318.Google Scholar
13. 12.
For instance, it appears that some commentators have misinterpreted Frege because he does not assume the definability interpretation. (For Frege, functions exist objectively independently of their representability in language.) From this they have in effect mistakenly inferred that Frege accepted the standard interpretation. See here Hintikka and Sandu, op. cit.Google Scholar
14. 13.
Jon Barwise and Solomon Feferman (eds.): 1986, Model-theoretical Logics, Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar
15. 14.
Quoted in Umberto Bottazzini, The “Higher Calculus”: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, Berlin-Heidelberg-New York, 1986, p. 33.Google Scholar
16. 15.
I. H. Anellis, A History of Mathematical Logic in Russia and the Soviet Union, unpublished.Google Scholar
17. 16.
See L. Euler: 1990, Introduction to Analysis of the Infinite, Book II,translated by John D. Blanton, Springer-Verlag, Berlin-Heidelberg-New York, p. 6, section 9.Google Scholar
18. 17.
See his paper, `Mathematical Ideas, Ideas, and Ideology’, The Mathematical Intelligencer 14(2)(Spring 1992), 6–19 (here p. 7b).Google Scholar
19. 18.
The first quotation is from Judith V. Grabiner: 1981, The Origins of Cauchy’s Rigorous Calculus, The MIT Press, Cambridge MA, pp. 89–90. The second is from Thomas Hawkins: 1970, Lebesgue’s Theory of Integration, University of Wisconsin Press, Madison, p. 4.Google Scholar
20. 19.
P. Dugac: 1973, “Eléments d’analyse de Karl Weierstrass”, Archive of the History of Exact Sciences 10,41–176. (See p. 71; quoted in Bottazzini, op. cit., p. 199.)Google Scholar
21. 20.
In Kronecker’s View of the Foundations of Mathematics, in David E. Rowe and John McCleary (eds.), The History of Modern Mathematics, vol. I, Academic Press, San Diego, pp. 67–77. (See here p. 74.)Google Scholar
22. 21.
For Weierstrass’s work, see Felix Klein: 1927, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, vol. 1, Springer-Verlag, Berlin-Heidelberg, pp. 276–295.Google Scholar
23. 22.
Leopold Kronecker: 1886, `Über einige Anwendungen der Modulsysteme auf elementare algebraische Fragen’, Journal für reine und angewandte Mathematik, vol. 99, pp. 329–371, especially p. 336. Quoted in Joseph W. Dauben: 1979, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard U.P., Cambridge MA, p. 68.Google Scholar
24. 23.
Michael Hallett: 1984, Cantorian Set Theory and Limitation of Size, Clarendon Press, Oxford.Google Scholar
25. 24.
See here Jaakko Hintikka: 1994, `What is Elementary Logic? Independence-friendly Logic as the True Core Area of Logic’, in K. Gavroglu et al. (eds.), Physics, Philosophy and Scientific Community: Essays in Honor of Robert S. Cohen, Kluwer Academic, Dordrecht, pp. 301–326.Google Scholar
26. 25.
See Stewart Shapiro: 1991, Foundations without Foundationalism, Clarendon Press, Oxford.Google Scholar
27. 26.
Bertrand Russell: 1908, `Mathematical Logic as Based on the Theory of Types’, American Journal of Mathematics, vol. 30, pp. 222–262, reprinted in Bertrand Russell: 1956, Logic and Knowledge: Essays 1901–1950, ed. by Robert C. Marsh, Allen and Unwin, London, pp. 59–102.Google Scholar
28. 27.
Bertrand Russell and Alfred North Whitehead: 1910–1913, Principia Mathematica 1-I11, Cambridge University Press, Cambridge; second ed., 1927.Google Scholar
29. 28.
Op. cit., note 26, second edition, vol. 1, Appendix B.Google Scholar
30. 29.
Frank P. Ramsey: 1925, `The Foundations of Mathematics’, Proceedings of the London Mathematical Society, Ser. 2, vol. 25, part 5, pp. 338–384. Reprinted (among other places) in F. P. Ramsey: 1978, Foundations, ed. by D. H. Mellor, Routledge and Kegan Paul, London, pp. 152–212.Google Scholar
31. 30.
Op. cit., p. 173 of the reprint.Google Scholar
32. 31.
Op. cit., p. 165 of the reprint.Google Scholar
33. 32.
See Maria Carla Galavotti (ed.): 1991, Frank Plumpton Ramsey, Notes on Philosophy, Probability and Mathematics, Bibliopolis, Napoli, Appendix, and Mathieu Marion’s contribution to the present volume.Google Scholar
34. 33.
For a discussion of the history of this contrast, see Gregory H. Moore: 1988, “The Emergence of First-Order Logic”, in William Aspray and Philip Kitcher (eds.), History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science, vol. II), University of Minnesota Press, Minneapolis, 1988, pp. 95–135.Google Scholar
35. 34.
See here Merrill B. Hintikka and Jaakko Hintikka: 1986, Investigating Wittgenstein, Basil Blackwell, Oxford, chapters 2 and 4.Google Scholar
36. 35.
Cf. Gregory H. Moore, op. cit., note 10 above.Google Scholar
37. 36.
Bertrand Russell: 1919, Introduction to Mathematical Philosophy, Allen and Unwin, London, chapter 12, especially p. 126.Google Scholar
38. 37.
39. 38.
See op. cit., note 5 above, and cf. Jaakko Hintikka: 1993, `Gödel’s Functional Interpretation in Perspective’, in M. D. Schwabl (ed.), Yearbook of the Kurt Gödel Society, Vienna, pp. 5–43.Google Scholar
40. 39.
See Hintikka, op. cit., note 3 above.Google Scholar
41. 40.
Michael Dummett, Elements of Intuitionism, Clarendon Press, Oxford, 1977, pp. 52–53 and 314.Google Scholar
42. 41.
Op. cit., note 10 above, especially pp. 64–76.Google Scholar
43. 42.
Op. cit., note 20 above, p. 71.Google Scholar
44. 43.
See here Jaakko Hintikka: 1988, `What Is the Logic of Experimental Inquiry?’, Synthese 74, 173–190.Google Scholar
45. 44.
For an early discussion of the nature and role of conclusiveness conditions, see Jaakko Hintikka, The Semantics of Questions and the Questions of Semantics (Acta Philosophica Fennica vol. 28, no. 4) Societas Philosophica Fennica, Helsinki, 1976, especially ch. 3. The analysis presented there is now being generalized, especially to questions whose answers are functions.Google Scholar