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Sequences

Published online by Cambridge University Press:  22 April 2025

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.

Information

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1941

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