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BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

In recent years, various kinds of circular phenomena have been studied by researchers from different disciplines. Examples are the study of automata in the science of computation, which generally contain cycles of state transitions; recursive domain equations in computer science, which play a role in the semantics of concurrent programming languages with recursion; theories of truth and (self-)reference in philosophy; and terminological cycles in artificial intelligence. Let us look at an example, taken from computation science, and treated in Vicious circles in all detail. Streams are infinite sequences of elements over an (alphabet) set A. Let a and b be in A and let s be the stream that informally is defined as consisting of an a followed by a b, again followed by an a and a b, and so on. Because of the fact that when the first two elements of s are removed, the same stream s is obtained again, it seems natural to model it by using the ordinary pairing operator: s = 〈a, 〈b, s〉〉, or, using t to denote the stream obtained by removing the first a from s, by the following system of equations: s = 〈a, t〉 and t = 〈b, s〉. This description is satisfactory in that it seems to capture our intuition well. However, it is inconsistent with the standard axioms of set theory. The most familiar universe of sets is the so-called iterative or cumulative universe. It is described by the axioms of basic set theory, ZFC−, to which the Foundation Axiom (FA) is added. This axiom states that the membership relation ∈ on any set is wellfounded. An equivalent formulation is that the universe of all sets consists of a union of layers, corresponding to the various stages of construction. Starting with the empty set, at each stage new sets are built from already existing ones by means of the set-theoretical operations. It follows that the universe determined by ZFC (= ZFC− + FA) does not contain sets Ω that satisfy Ω = {Ω}. The stream s of our example above does not belong to this universe either: In the construction of s, the stream t is used which, at its turn, is constructed using s. This kind of circle is in the universe of ZFC to be considered ‘vicious’.