4 problems

Let ${\sf iPH}_{\Sigma}$ be the logic introduced in the subsection for $\Sigma_1$-preservativity, and let a $\Sigma_1$-substitution $\sigma$ assign $\Sigma_1$ arithmetical formulas…

Let ${\sf iPH}$ be the logic over the language with binary modal operator $\rhd$ introduced for arithmetical interpretations in Heyting arithmetic ${\rm HA}$. An arithmetical inter…

Kleene's realizability conjecture. Every closed theorem of $\mathrm{HA}$ is realizable.

Let ${\sf HA}^*$ be the arithmetic theory considered in the paper, and let $\Box A$ denote the provability modality for modal propositions. Let $\iH^*$ be the intuitionistic modal…