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A minimalist two-level foundation for constructive mathematics

Authors
Journal
Annals of Pure and Applied Logic
0168-0072
Publisher
Elsevier
Publication Date
Volume
160
Issue
3
Identifiers
DOI: 10.1016/j.apal.2009.01.006
Keywords
  • Intuitionistic Logic
  • Set Theory
  • Type Theory
Disciplines
  • Computer Science
  • Logic
  • Mathematics

Abstract

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 a previous version 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.

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