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

Abstract

In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnas' limit definition of these within Martin-Lof's theory of types. A system is a type $W$ together with an assignment of $\bar{\alpha} \in U$ and $\tilde{\alpha} \in \bar{\alpha} \rightarrow W$ to each $\alpha \in W$. We show that for any system $W$ we can define an equivalence relation $=_w$ such that $\alpha =_w \beta \in U$ 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 $(\ast)$, then we can interpret CZF minus the set induction scheme in $W$. $W$ is then extended to a complete system $W^\ast$ by taking limits of approximation chains. We show that in $W^\ast$ the antifoundation axiom AFA holds as well as the axioms of $\mathrm{CFZ}^-$.