Sets in Types, Types in Sets

  title={Sets in Types, Types in Sets},
  author={Benjamin Werner},
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.
Large model constructions for second-order ZF in dependent type theory
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.
On Relating Type Theories and Set Theories
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
It is proved that Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the categricity of extended axiomatisations fixing the number of Grothendieck universes are correct.
Formalised Set Theory : Well-Orderings and the Axiom of Choice
In this thesis, we give a substantial formalisation of classical set theory in the proof system Coq. We assume an axiomatisation of ZF and present a development of the theory containing relations,
From type theory to setoids and back
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
lamda-Z: Zermelo's Set Theory as a PTS with 4 Sorts
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
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.


Constructions, inductive types and strong normalization
An extension of the core calculus by inductive types is investigated and it is shown how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive type types.
Inductive sets and families in Martin-Lo¨f's type theory and their set-theoretic semantics
Martin-LL of's type theory is presented in several steps. The kernel is a dependently typed-calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive
An extended calculus of constructions
This thesis presents and studies a unifying theory of dependent types ECC | Extended Calculus of Constructions, a strong and expressive calculus for formalization of mathematics, structured proof development and program speci cation, and shows the proof-theoretic consistency of the calculus.
On computational open-endedness in Martin-Lof's type theory
  • Douglas J. Howe
  • Computer Science
    [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
  • 1991
The main result is the definition of a particular computation system that includes a collection of oracles sufficient to provide a classical semantics for Martin-Lof's type theory in which the excluded middle law holds.
The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles
Intuitionistic Sets and Ordinals
It is shown how to make the successor monotone by introducing plumpness , which strengthens transitivity, and clarifies the traditional development of successors and unions, making it intuitionistic ; even the (classical) proof of trichotomy is made simpler.
ECC, an extended calculus of constructions
  • Zhaohui Luo
  • Mathematics
    [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science
  • 1989
A higher-order calculus ECC (extended calculus of constructions) is presented which can be seen as an extension of the calculus of constructions by adding strong sum types and a fully cumulative type
A Formulation of the Simple Theory of Types
A formulation of the simple theory oftypes which incorporates certain features of the calculus of λ-conversion into the theory of types and is offered as being of interest on this basis.
A Generic Normalisation Proof for Pure Type Systems
We prove the strong normalisation for any PTS, provided the existence of a certain Λ-set \(\mathfrak{A}^ \Uparrow \) (s) for every sort s of the system. The properties verified by the \(\mathfrak{A}^