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Sequences

Published online by Cambridge University Press:  22 April 2025

Nelson Goodman
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Extract

The problem of defining ordered couples on the basis of class theory, and thus reducing relation theory to class theory, was first solved by Wiener. By Kuratowski's somewhat simpler device, the ordered couple x,y is defined as

The principle involved may be extended for sequences of any finite length. The ordered triple x,y,z would, for example, become:

By this procedure any sequence of length n would be defined as a class n types higher than the components of the sequence; and there appears to be no alternative extension of Kuratowski's device that would escape this progressive raising of type. Thus a new method of defining sequences that would construe them as classes only one type higher than their components would effect a considerable simplification in the theory of sequences and relations. My purpose in this paper is to propose and explain such a definition and some of its consequences.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 6 , Issue 4 , December 1941 , pp. 150 - 153
Copyright
Copyright © Association for Symbolic Logic 1941

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References

1 N. Wiener, A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, vol. 17 (1914), pp. 387-390.

2 C. Kuratowski, Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta malhematicae, vol. 2 (1921), pp. 161-171.

3 That defining the ordered triple x,y,z as iix U i(ix U iy) U i(ix u iy u iz) would fail to distinguish the sequences x,x,y and x,y,x and x,y,y from each other has been pointed out by W. V. Quine.

4 I hope in a later paper to deal with certain consequences for the measurement of the logical economy of bases for extralogical systems.

5 In Mathematical logic (New York 1940, p. 202, fine type), Quine suggested an alternative to the definition of ordered couple he used in his system, but later discovered that this alternative definition would fail to distinguish x,y from y,x when x⊂y. In a later discussion between us he proposed the first satisfactory definition of the ordered couple as a class only one type higher than its components. However his device—which defines x,y as the class of the unit subclasses of x and the complements of unit subclasses of y— is obviously not readily extensible to longer sequences. It has the possible further disadvantages, as compared with the definition I suggest above, (1) that in an infinite universe an ordered couple of finite components will always have infinite classes among its members, and (2) that the couple A,A will not be identified with the null class.

6 This is done in Quine's Mathematical logic, but is compatible with systems involving type theory if we merely modify type theory by regarding individuals, and the null class as well, as belonging to all types. Every class of two or more individuals still belongs to type 1 only; every class having as members some classes that belong only to type 1 belongs to type 2 only, and so on. Incidentally, adoption of this point of view makes a sequence of length 1, provided its first component is an individual or class of individuals, identical under our definition with that component. Further, no sequence of individuals is of higher type than its components; it is for this reason that I write “not more than” rather than “only” in the first and last sentences of the paragraph following.

7 A study of qualities (Doctoral thesis, Harvard Library, 1941), p. 44.

8 Throughout it should be remembered that the components of a sequence may be quite different from the classes belonging to it as members.

9 Since seeing the above definitions, Quine has suggested that sequences might be handled in an alternative way, defining x1,x2,…,xn where n>2 as ŷž(y∊x1 · z∊1 · v · yex1 · z∊2 · v · … · v · y∊xn · z∊n). A component xk of a sequence Q would then be defined as Q11k. The simplicity of this method is rather offset by the fact that it rests upon prior definition of the ordered couple, so that two steps are involved. It is questionable whether the formal definitions corresponding to these schemata would be notably simpler than those I give above. Moreover, the interesting consequences concerning the relation of all classes to sequences (dealt with in §2 of the present paper) would not be forthcoming.

10 Were it not for the identification of individuals with their unit classes, we should have to read “class of classes” here and in similar contexts. However, when this identifiestion is made (as outlined in footnote 6 above), even a class of individuals is a class of classes, i.e., of unit classes.