Axiomatic Justification in Constructive Morse Set Theory

Abstract

Working within Constructive Morse Set Theory (CMST), we introduce axioms for a new notion, jst Pp, intended to capture what it means for P to prove, or justify, p under the BHK interpretation of intuitionistic logic. Since it makes no distinction between terms and formulae -- every term is also a formula, and vice versa -- CMST is well suited to our axiomatic development of justification theory within set theory itself. After stating our axioms for jst Pp, we derive many consequences thereof. In particular, we show that (with certain restrictions) our axioms for jst Pp align with the intended BHK interpretations of the axioms of intuitionistic logic.