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A Construction of Non-Well-Founded Sets within Martin-Löf's Type Theory

Ingrid Lindström
The Journal of Symbolic Logic
Vol. 54, No. 1 (Mar., 1989), pp. 57-64
DOI: 10.2307/2275015
Stable URL: http://www.jstor.org/stable/2275015
Page Count: 8
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A Construction of Non-Well-Founded Sets within Martin-Löf's Type Theory
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Abstract

In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löf's theory of types. A system is a type W together with an assignment of ᾱ ∈ U and α̃ ∈ ᾱ → W to each α ∈ W. We show that for any system W we can define an equivalence relation =w such that α =w β ∈ 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-.

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