A minimalist two-level foundation for constructive mathematics

TL;DR

This paper introduces a minimal two-level foundation for constructive mathematics, combining an intensional type theory with an extensional set theory interpreted via quotient models, emphasizing minimality and constructiveness.

Contribution

It proposes a novel minimal two-level framework linking intensional and extensional theories for constructive mathematics, enhancing foundational simplicity and computational interpretation.

Findings

Two-level theory effectively formalizes constructive mathematics.

Extensional level is interpreted within intensional type theory using quotient models.

Framework maintains minimality while supporting proofs-as-programs paradigm.

Abstract

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in the mentioned paper with collections. The other level is given by an extensional set theory which 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.