Extending Type Theory with Forcing

International audienceThis paper presents an intuitionistic forcing translation for the Calculus of Constructions (CoC), a translation that corresponds to an internalization of the presheaf construction in CoC. Depending on the chosen set of forcing conditions, the resulting type system can be extended with extra logical principles. The translation is proven correct-in the sense that it preserves type checking-and has been implemented in Coq. As a case study, we show how the forcing translation on integers (which corresponds to the internalization of the topos of trees) allows us to define general inductive types in Coq, without the strict positivity condition. Using such general inductive types, we can construct a shallow embedding of the pure \lambda-calculus in Coq, without defining an axiom on the existence of an universal domain. We also build another forcing layer where we prove the negation of the continuum hypothesis

Is the dream solution to the continuum hypothesis attainable?

The dream solution of the continuum hypothesis (CH) would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to CH is unattainable. The article is adapted from and expands upon material in my article, "The set-theoretic multiverse", to appear in the Review of Symbolic Logic (see arXiv:1108.4223).Comment: This article is based upon an argument I gave during the course of a three-lecture tutorial on set-theoretic geology at the summer school "Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Developments", at the University of London, Birkbeck in August 201

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

Forcing consequences of PFA together with the continuum large

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.Comment: 35 page