A Negationless Interpretation of Intuitionistic Axiomatic Theories: Arithmetic and Analysis

The present work contains an axiomatic treatment of some parts of the version of intuitionistic mathematics advocated by G.F.C. Griss, the so-called \negationless intuitionistic mathematics" (see, e. Formal systems NPC, NA, and FIM N for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 6 in Section 2 asserts the translatability of Heyting's arithmetic HA into NA. This result can in fact be extended to a large class of intuitionistic theories based on HA and their negationless counterparts. For instance, in Section 3 this is shown for Kleene's system of intuitionistic analysis FIM and our FIM N .