The Russell Operator

Constructivist Foundations 7 (2):112-115 (2012)

Louis H. Kauffman
University of Illinois at Chicago
Context: The question of how to understand the epistemology of set theory has been a longstanding problem in the foundations of mathematics since Cantor formulated the theory in the 19th century, and particularly since Bertrand Russell articulated his paradox in the early twentieth century. The theory of types pioneered by Russell and Whitehead was simplified by mathematicians to a single distinction between sets and classes. The question of the meaning of this distinction and its necessity still remains open. Problem: I am concerned with the meaning of the set/class distinction and I wish to show that it arises naturally due to the nature of the sort of distinctions that sets create. Method: The method of the paper is to discuss first the Russell paradox and the arguments of Cantor that preceded it. Then we point out that the Russell set of all sets that are not members of themselves can be replaced by the Russell operator R, which is applied to a set S to form R(S), the set of all sets in S that are not members of themselves. Results: The key point about R(S) is that it is well-defined in terms of S, and R(S) cannot be a member of S. Thus any set, even the simplest one, is incomplete. This provides the solution to the problem that I have posed. It shows that the distinction between sets and classes is natural and necessary. Implications: While we have shown that the distinction between sets and classes is natural and necessary, this can only be the beginning from the point of view of epistemology. It is we who will create further distinctions. And it is up to us to maintain these distinctions, or to allow them to coalesce. Constructivist content: I argue in favor of a constructivist perspective for set theory, mathematics, and the way these structures fit into our natural language and constructed speech and worlds. That is the point of this paper. It is only in the reach for absolutes, ignoring the fact that we are the authors of these structures, that the paradoxes arise
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