A $j$-translation with Kripke forcing relation

In this paper, we introduce a translation that combines the $j$-translation with Kripke forcing in the internal logic of an elementary topos. First, we show that our translation is sound for intuitionistic first-order logic and Heyting arithmetic. Furthermore, its interpretation in the effective topos provides an extension of the sheaf model of realizability introduced by de Jongh and Goodman. As an application, we systematically investigate translations for semi-classical axioms. Based on this investigation, we establish a separation result on semi-classical arithmetics, which cannot be obtained using the usual $j$-realizability.

💡 Research Summary

The paper introduces a novel translation that merges the j‑translation with Kripke forcing inside the internal logic of an elementary topos. After reviewing the internal language of a topos and the notion of a local operator (an internal nucleus on the subobject classifier Ω), the author defines a variable‑indexed family of local operators and a sub‑poset P of such operators. The new translation, denoted j ⊩ P φ, first applies the usual internal j‑translation to a formula φ (adding j to atomic predicates, ∨, and ∃) and then requires that the resulting formula be forced by every operator in P.

The main technical results are soundness theorems: the translation preserves truth for intuitionistic first‑order logic (IQC) and for Heyting arithmetic (HA). The proofs are carried out entirely within the higher‑order internal language, using basic properties of nuclei (Lemma 3.3) to show that the added j‑closures do not break logical inference rules.

In the effective topos Eff, each partial recursive function f gives rise to a local operator j_f, and a collection T of partial functions yields a sub‑poset P. The author defines “P‑realizability” (Definition 5.9) and proves that it extends the de Jongh‑Goodman realizability (Theorem 5.20). Thus P‑realizability can be seen as Kleene realizability relative to an oracle together with a Kripke forcing layer.

The paper then systematically translates several semi‑classical principles (Σ_{n+1}‑DNE, Π_{n+1}‑DNE, and their double‑negated variants). By analysing how these principles behave under the new translation, the author establishes a separation theorem (Theorem 6.19): over HA, the schema Σ_{n+1}‑DNE + ¬¬(Π_{n+1} ∨ Π_{n+1}‑DNE) does not imply Π_{n+1} ∨ Π_{n+1}‑DNE. This separation cannot be obtained with ordinary j‑realizability, highlighting the extra discriminating power of P‑realizability.

Overall, the work provides a unified framework that generalises the j‑translation by allowing quantification over internal local operators, connects it to realizability via the effective topos, and applies it to obtain new proof‑theoretic separation results for semi‑classical arithmetic. Future directions suggested include extending the method to other semi‑classical axioms, choice principles, and higher‑type settings.