The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics.Formal systems NPC, NA, and FIMN for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section (...) 3 this is shown for Kleene's system of intuitionistic analysis FIMand our FIMN. (shrink)

This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Gödel's completeness theorem for classical predicate logic.

The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics.Formal systems NPC, NA, and FIMN for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section (...) 3 this is shown for Kleene's system of intuitionistic analysis FIMand our FIMN. (shrink)

In a seriesof papers beginning in 1944, the Dutch mathematician and philosopherGeorge Francois Cornelis Griss proposed that constructivemathematics should be developedwithout the use of the intuitionistic negation1 and,moreover, without any use of a nullpredicate.In the present work, we give formalized versions of intuitionisticarithmetic, analysis,and higher-order arithmetic in the spirit ofGriss' ``negationless intuitionistic mathematics''and then consider their relation to thecurrent formalizations of thesetheories.

This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.

This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.

In a seriesof papers beginning in 1944, the Dutch mathematician and philosopherGeorge Francois Cornelis Griss proposed that constructivemathematics should be developedwithout the use of the intuitionistic negation1 and,moreover, without any use of a nullpredicate.In the present work, we give formalized versions of intuitionisticarithmetic, analysis,and higher-order arithmetic in the spirit ofGriss' ``negationless intuitionistic mathematics''and then consider their relation to thecurrent formalizations of thesetheories.

Let HA be Heyting's arithmetic, and let CS denote the conjunction of Kreisel's axioms for the creative subject: \begin{eqnarray*} {\rm CS}_1.&&\quad \,\forall\, x (\qed_x A \vee \; \neg \qed_x A)\; ,\nn {\rm CS}_2. &&\quad \,\forall\, x (\qed_x A\to A)\; ,\nn {\rm CS}_3^{\rm S}. &&\quad A\to\,\exists\, x \qed_x A\; ,\nn {\rm CS}_4.&&\quad \,\forall\, x\,\forall\, y (\qed_x A & y \ge x\to\qed_y A)\; .\nn \end{eqnarray*} It is shown that the theory HA + CS with the induction schema restricted to arithmetical (i.e. not (...) containing $\qed$ ) formulas is conservative over HA. (shrink)

Within a weak system \ of intuitionistic analysis one may prove, using the Weak Fan Theorem as an additional axiom, a completeness theorem for intuitionistic first-order predicate logic relative to validity in generalized Beth models as well as a completeness theorem for classical first-order predicate logic relative to validity in intuitionistic structures. Conversely, each of these theorems implies over \ the Weak Fan Theorem.