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Sets, Aggregates, and Numbers

Published online by Cambridge University Press:  01 January 2020

Palle Yourgrau
Palle Yourgrau*
Affiliation:
University of Wisconsin-Madison, 600 North Park Street, Madison, WS53706
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Extract

Frege's definition of the natural number n in terms of the set of n-membered sets has been treated rudely by history. It has suffered not one but two crippling blows. The discovery of Russell's Paradox revealed a fatal flaw in the ‘naive’ conception of set. In spite of its intuitive appeal, Frege's Basic Law V (in the context of the rest of his theory) turned out to be impermissible, leaving us only with the etiolated concept of set that survives in the axiomatic treatments initiated by Zermelo. The independence results, however, of Godel and Cohen, concerning Cantor's Continuum Hypothesis, have left us adrift in choosing between Cantorian and non-Cantorian set theories, which has induced in some logicians a skepticism in regard to the very idea of set-theoretic platonism.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 15 , Issue 4 , December 1985 , pp. 581 - 592
Copyright
Copyright © The Authors 1985

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References

1 For an excellent discussion of these points see Giaquinto, Marcus ‘Hilbert's Philosophy of Mathematics,’ British Journal for the Philosophy of Science , 34 (1983) 119–32.CrossRefGoogle Scholar

2 See his ‘What is Cantor's Continuum Hypothesis?, ‘in Benacerraf and Putnam, eds., Philosophy of Mathematics (Cambridge, MA: Cambridge University Press Second Edition 1983) 470-85.

3 See his ‘Russell's Mathematical Logic’ in Benacerraf and Putnam, eds., 449.

4 See Rapaport, William J. a) ‘Meinongian Theories and a Russellian Paradox,’ Noûs, 12 (1978) 153–80;CrossRefGoogle Scholar b) ‘How to Make the World Fit Our Language,’ Grazer Philosophische Studien , 14 (1981) 1-21 and Byrd, Michael ‘Russell's Paradox, Sense and Reference,’ Xerox , Department of Philosophy, University of Wisconsin-Madison.Google Scholar

5 I would suggest, rather, that the real source of trouble here concerns when definite descriptions denote. Frege's original troubles with Basic Law V turn on a version of the principle: F(1X(Fx)). (Substitute for ‘Fx’ : ‘(y) (yεx .≡. Fy)’). But this is not the place to pursue this issue.

6 ‘What Numbers Could Not Be,’ in Benacerraf and Putnam, eds., 272-94

7 ‘Set-Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations,’ The Canadian journal of Philosophy , 12 (1982) 353-74

8 See, for example, Word and Object (Cambridge, MA: Technology Press of the Massachusetts Institute of Technology 1960).

9 Noûs 15 (1981) 495-513

10 To be precise, as Prof. Maddy has pointed out to me, one should here speak rather ot the proper class of n-membered ‘collections’ (since proper classes themselves will be included).

11 See Burge, Tyler ‘Frege on Extensions of Concepts, from 1884 to 1903,’ The Philosophical Review 93 (1984) 3–34.CrossRefGoogle Scholar

12 For a useful discussion of this issue, see Wright, Crispin Frege's Conception of Numbers as Objects (Aberdeen: Aberdeen University Press 1983).Google Scholar

13 See Charles Parsons, ‘What is the Iterative Concept of Set?,’ in Benacerraf and Putnam, eds., 503-29.

14 ‘Mathematical Truth,’ in Benacerraf and Putnam, eds., 403-20

15 I have expressed my misgivings about the causal or ‘new theory of reference’ in ‘Frege, Perry, and Demonstratives,’ The Canadian journal of Philosophy , 12 (1982) 725-53, and concerning the issue of mathematical knowledge in particular, in ‘Informational Retrieval and Cognitive Accessibility,’ forthcoming in Barry Loewer, ed., Synthese: Information-Semantics and Epistemology.

16 ‘Perception and Mathematical Intuition, ’The Philosophical Review , 89 (1980) 163-96

17 As Ellery Eells has reminded me.

18 ‘Frege, Mill and the Foundations of Arithmetic,’ Journal of Philosophy , 77 (1980) 65-80

19 Kessler, 68

20 The Philosophical Review,89 (1980) 607-24

21 ‘Against the Aggregate Theory of Number,’ Journal of Philosophy , 79 (1982) 163-68

22 Compare with Quine's discussion of the concept of identity in his ‘Review of Munitz, M. ed., Identity and Individuation,,’ Journal of Philosophy , 69 (1972) 488–97.Google Scholar He observes that the puzzle cases about personal identity raise questions about the correct analysis of persons, but should not give rise to queries concerning the nature of identity.