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A construction of non-well-founded sets within Martin-Löf's type theory

Published online by Cambridge University Press:  12 March 2014

Ingrid Lindström
Ingrid Lindström*
Affiliation:
Matematiska Institutionen, Uppsala Universitet, 752 38 Uppsala, Sweden
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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-Löfs theory of types. A system is a type W together with an assignment of and to each ∝ ∈ W. We show that for any system W we can define an equivalence relation =w such that ∝ =w ß ∈ 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−.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 57 - 64
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

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