# Sets in Types, Types in Sets

@inproceedings{Werner1997SetsIT,
title={Sets in Types, Types in Sets},
author={Benjamin Werner},
booktitle={TACS},
year={1997}
}
We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Fraenkel set theory and the opposite way. More precisely, we actually construct two families of encodings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r.t. expressive power and thus are essentially equivalent. Both encodings are quite elementary: type…
A Normalizing Intuitionistic Set Theory with Inaccessible Sets
An impredicative constructive version of Zermelo-Fraenkel set theory IZF with Replacement and $\omega$-many inaccessibles is axiomatized, which allows a weakly-normalizing typed lambda calculus corresponding to proofs in \izfio according to the Curry-Howard isomorphism principle.
Sets in Coq, Coq in Sets
The long-term goal is to build a formal set theoretical model of the Calculus of Inductive Constructions, so the authors can be sure that Coq is consistent with the language used by most mathematicians.
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It is shown that the consistency strength of Coq extended by excluded middle and a description operator on well-founded trees allows for constructing models with exactly n Grothendieck universes for every natural number n.
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An easy way to determine an upper bound on the proof theoretic strength of the type theories implemented in the proof development systems Lego and Coq is to use the ‘obvious’ types-as-sets interpretation of these type theories in a strong enough classical axiomatic set theory.
Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory
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Journal of Automated Reasoning
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A model of Martin-Lof extensional type theory with universes is formalized in Agda, an interactive proof system based on Martin-Lof intensional type theory. This may be understood, we claim, as a
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It is shown that the embedding of the language of set theory into λZ via the ‘sets as pointed graphs' translation makesλZ a conservative extension of IZ+AFA+TC (intuitionistic Zermelo's set theory plus Aczel's antifoundation axiom plus the axiom of transitive closure)—a theory which is equiconsistent to Zermello's.
CoQMTU: A Higher-Order Type Theory with a Predicative Hierarchy of Universes Parametrized by a Decidable First-Order Theory
• Mathematics
2011 IEEE 26th Annual Symposium on Logic in Computer Science
• 2011
It is shown that CoqMTU enjoys all basic meta-theoretical properties of such calculi, confluence, subject reduction and strong normalization when restricted to weak-elimination, implying the decidability of type-checking in this case as well as consistency.
Surreal Numbers in Coq
This paper discusses in particular the definitional or proving points where I had to diverge from Conway's or the most natural way, like separation of simultaneous induction-recursion into two inductions, transforming the definition of the order into a mutually inductive definition of “at most” and ‘at least’ and fitting the rather complicated induction/recursion schemes into the type theory of Coq.

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