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Proper classes

Published online by Cambridge University Press:  12 March 2014

Penelope Maddy
Penelope Maddy*
Affiliation:
University of Notre Dame, Notre Dame, Indiana 46556
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Extract

The modern problem of infinity was first raised by Aristotle who held (at least on the popular interpretation) that infinite sets exist potentially (i.e. one more number can always be counted, one more division can always be made in a line segment) but not actually (i.e. the numbers or divisions cannot all exist at one time). In fact, Aristotle not only held that completed infinities never actually exist, but also that they are impossible, that is, that the assumption that they do exist leads to contradictions. To see this, consider Aristotle's view that mathematical entities depend for their existence on the existence of primary substance in which they inhere, coupled with his view that there can be no infinite body. From these it follows that if there were actual completed infinities, infinite bodies would both exist and not exist. The details here are not so important as the idea that if a completed infinite is assumed to exist a contradiction follows.

This negative attitude towards completed infinities flourished for centuries. By the mid-1500's, the German mathematician Stifel (who apparently also invented Pascal's triangle) was moved to condemn irrational numbers simply by their association with the completed infinite:

… just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity.

About a hundred years later, Galileo further discredited the completed infinite by pointing out that line segments of different lengths can be brought into one-to-one correspondence by projection, as can the natural numbers and the perfect squares by assigning each number to its square.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 1 , March 1983 , pp. 113 - 139
Copyright
Copyright © Association for Symbolic Logic 1983

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References

2 As often happens, the popular interpretation misses numerous subtleties and controversies. For example, see Hintikka, J., Aristotelian infinity, Philosophical Review, vol. 75 (1966), pp. 197–218CrossRefGoogle Scholar, and Lear, J., Aristotelian infinity, Proceedings of the Aristotelian Society, vol. 80, pp. 187–210CrossRefGoogle Scholar.

3 Quoted in Kline, M., Mathematical thought from ancient to modern times, Oxford, 1972, p. 251Google Scholar.

4 The analyst, paragraph 14, The works of George Berkeley, Bishop of Cloyne (Luce, A. and Jessop, T., eds.), Edinburgh, 1948–1957Google Scholar.

5 For example, the two sets of rationals which make up a Dedekind cut are both infinite, yet they are considered sufficiently complete to be combined into ordered pairs, then infinite sets of such ordered pairs, and so on.

6 See Jourdain's, P. introduction to Cantor's, Contributions to the founding of the theory of transfinite numbers, Dover, New York, pp. 67–68Google Scholar, and Dauben, J., George Cantor, Harvard, 1979, pp. 132–133 and the references cited thereGoogle Scholar.

7 Gödel's mathematical realism is a more developed form of this view. See his Russell's mathematical logic and What is Cantor's continuum problem? in Philosophy of mathematics (Benacerraf, P. and Putnam, H., eds.), Prentice-Hall, Princeton, N.J., 1964, pp. 211–232, 258–273Google Scholar. I have presented and defended a descendant of Gödel's view in Perception and mathematical intuition, Philosophical Review, vol. 89 (1980), pp. 163–196CrossRefGoogle Scholar, and in Sets and numbers, Nous, vol. 15 (1981), pp. 495–511CrossRefGoogle Scholar. Both derive from my doctoral dissertation, Set theoretic realism, Princeton University, 1979Google Scholar.

8 I say “more or less” because this is really Cantor's argument that the set of all ordinal numbers cannot exist. Then he shows that there are as many cardinals as ordinals, and hence that the set of all cardinals cannot exist. See Cantor's, Letter to Dedekind in Mathematical logic from Frege to Gödel. (van Heijenoort, J., ed.), Harvard University Press, Cambridge, Mass., 1967, pp. 113–117Google Scholar.

9 Cantor, , Letter to Dedekind, p. 114Google Scholar.

10 In common usage, a proper class is a collection which for some reason or other cannot be a set. Later I will take up the problem of characterizing the difference between classes generally and sets. Then a proper class will be a class which is not coextensive with any set.

11 From a letter dated 20 June, 1908 to Grace Chisholm Young, an English mathematician. Cited in Dauben, J., George Cantor, Isis, vol. 69 (1978), p. 547Google Scholar.

12 For a more complete discussion of Cantor's interactions with the church and how they were influenced by Leo XII's encyclical Aeterni Patris of 1879, see Dauben, J., George Cantor, pp. 140–13OGoogle Scholar

13 One begins with V, the class of all sets, then performs a complicated construction using the measurable cardinal. See Drake, F., Set theory, North-Holland, Amsterdam, 1974, Chapter 6, Section 2Google Scholar.

14 One argues that V, the class of all sets, is “structurally indefinable”, and thus that any structural property of Khas to be shared by some set. These discussions sometimes involve Q, the class of all ordinals, and even such things as Q + 1, and Q + 2. See Reinhardt, W., Remarks on reflection principles, large cardinals and elementary embeddings, Axiomatic set theory (Jech, T., ed.), American Mathematical Society, Providence, R. I., 1974, pp. 189–205CrossRefGoogle Scholar.

15 See Benacerraf, P., What numbers could not be, Philosophical Review, vol. 74 (1965), pp. 47–73CrossRefGoogle Scholar. I discuss how proper classes fit into this problem in Sets and numbers.

16 This paper was a sequel to another flawed attempt to disprove the continuum hypothesis which König delivered to the Third International Congress of Mathematicians in 1904. This speech caused Cantor considerable unhappiness and may have contributed to one of his notorious breakdowns. See Dauben, J., George Cantor, pp. 543–544Google Scholar. The paper under discussion in the text is On the foundations of set theory and the continuum problem in van Heijenoort, op. cit., pp. 145–149.

17 Ibid., p. 148.

18 On Platonism in mathematics in P. Benacerraf and H. Putnam, op. cit., pp. 275–276.

19 The first explicit statement of this conception is probably in Zermelo, E., Uber Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47CrossRefGoogle Scholar. See also Kreisel, G., Two notes on the foundations of set theory, Dialectica, vol. 23 (1969), pp. 93–114CrossRefGoogle Scholar, and Boolos, George, The iterative conception of set, Journal of Philosophy, vol. 68 (1971), pp. 215–231CrossRefGoogle Scholar.

20 The temporal metaphor is considered inessential. See Parsons, C., What is the iterative conception of set? in Logic, foundations of mathematics and computability theory (Butts, and Hintikka, , eds.), Reidel, Dordrecht, 1977, pp. 335–367CrossRefGoogle Scholar.

21 D. Martin, Sets versus classes, circulated xerox. This was also noted by Bernays, op. cit., p. 276.

22 What is Cantor's continuum problem?, pp. 262–263.

23 Parsons, C., Sets and classes, Nous, vol. 8 (1974), pp. 7–9CrossRefGoogle Scholar.

24 Martin, op. cit., pp. 9, 11.

25 There is considerable difficulty involved in simply seeing exactly what Russell's system is, what his ontology includes, and whether or not it does the job it is supposed to do. See Gödel's Russell's mathematical logic and Quine, W., Whitehead and the rise of modern logic in The philosophy of Alfred North Whitehead (Schilipp, P., ed.), Northwestern Universty Press, Evanston, 1941, pp. 127–163Google Scholar.

26 For discussion, see Fraenkel, A., Bar-Hillel, Y., and Levy, A., Foundations of set theory, North-Holland, Amsterdam, 1973, pp. 161–171Google Scholar.

27 An axiomatization of set theory in van Heijenoort, op. cit., pp. 396.

28 Ibid., p. 402.

29 A system of axiomatic set theory, this Journal, vol. 2 (1937), p. 65Google Scholar.

30 The consistency of the continuum hypothesis, Princeton University Press, Princeton, N. J., 1940, p. 2Google Scholar.

31 F. Drake, op. cit., p. 9.

32 Ackermann, W., Zur Axiomatik der Mengenlehre, Mathematische Annalen, vol. 131 (1956), pp. 336–345CrossRefGoogle Scholar. See also Levy, A., On Ackermann's set theory, this Journal, vol. 24 (1959), pp. 154–166Google Scholar.

33 This point is made in A. Fraenkel, Y. Bar-Hillel, and A. Levy, op. cit., p. 150.

34 See Grewe, R., Natural models of Ackermann's set theory, this Journal, vol. 34 (1969), pp. 481–488Google Scholar.

35 See Reinhardt, W., Set existence principles of Shoenfield, Ackermann, and Powell, Fundamenta Mathematicae, vol. 84 (1974), pp. 5–34CrossRefGoogle Scholar.

36 See C. Parsons, Sets and classes, and What is the iterative conception of set?

37 Lear, J., Sets and semantics, Journal of Philosophy, vol. 74 (1977), pp. 86–102CrossRefGoogle Scholar.

38 What is the iterative conception of set?, p. 350.

39 Ibid., p. 351.

40 Ibid., p. 355.

41 A similar formulation of the central difficulty appears in Rucker, R., The one I many problem in the foundations of set theory, Logic Colloquium 76, (Grandy, R. and Hyland, M., eds.), North-Holland, Amsterdam, 1977, pp. 567–593Google Scholar.

42 For discussion of this analogy see Parsons, C., The liar paradox, Journal of Philosophical Logic, vol. 3 (1974), pp. 381–412CrossRefGoogle Scholar.

43 Kripke, S., Outline of a theory of truth, Journal of Philosophy, vol. 72 (1975), pp. 690–716CrossRefGoogle Scholar.

44 See Frege, G., The basic laws of arithmetic, Universiiy of California Press, Berkeley and Los Angeles, California, 1967, translated and edited by Furth, M.. p. 139Google Scholar.

45 Russell's mathematical logic, p. 229.

46 Sets versus classes, p. 9.