JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. Abstract. In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnis' limit definition of these within Martin-LUf's theory of types. A system is a type W together with an assignment of a E U and a e Y -E W to each a E W. We show that for any system W we can define an equivalence relation =w such that a =w E e U and =w is the maximal bisimulation. Aczel's proof that CZF can be interpreted in the type V of iterative sets shows that if the system W satisfies an additional condition (*), then we can interpret CZF minus the set induction scheme in W. W is then extended to a complete system W* by taking limits of approximation chains. We show that in W* the antifoundation axiom AFA holds as well as the axioms of CFZ-.
?1. Introduction. In this paper, we give a constructive interpretation of non-wellfounded sets by formalizing Hallnis' definition of these, as expressed in [4], within Martin-Ldf's theory of types (see [5]). In [1] and [2], Aczel extended type theory to include a type V of iterative sets, and, within that extension, he gave an interpretation of the axiom system CZF for constructive set theory. The underlying logic for CZF is intuitionistic predicate logic. The main differences between the settheoretic axioms of CZF and ZF is that the separation axiom is assumed only for Aoformulas (and, therefore, CZF has a collection scheme rather than replacement) and that the power set axiom has been replaced by a weaker subset collection scheme. Also, CZF uses the set induction scheme rather than the axiom of foundation. CZF with classical logic is equivalent to ZF. Let CZF-denote CZF without the set induction scheme. Aczel's proof that CZF-can be interpreted in V yields an interpretation of CZF-in the system V* of non-well-founded sets constructed here, after that a suitable equality relation =v* on V* has been defined.
It then remains to show that we get an interpretation of the antifoundation axiom AFA. In Aczel's formulation (see [3]), AFA states that-every graph has a unique decoration. Here, a graph consists of a set G of nodes and a set E c G x G of edges, and a decoration of a graph is a function d that assigns a set da to each node a such that da{db I (a, b) E E}.Hallnds' concept of a non-well-founded set can be characterized as follows. Let W denote the universe of non-well-founded sets.
INGRID LINDSTROM (a) A non-well-founded set is the (projective) limit of an approximation chain of well-founded sets. In this sense, a non-well-founded set is an infinite object, constructed from well-founded ones. (b...
