We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05] . One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the "proofs-as-programs" paradigm and acts as a programming language.
The extensional level is modelled REF into the minimal theory using a quotient construction.
A minimalist two-level foundation for constructive mathematics
arXiv: Logic
Computer Science Mathematics