Higher-order illative combinatory logic

TL;DR

This paper constructs a model for higher-order illative combinatory logic, proving its consistency and embedding first-order intuitionistic logic into a related system, advancing understanding of logical systems' foundations.

Contribution

It provides a model construction for $ ext{I}_ ext{omega}$ and embeds first-order intuitionistic logic into $ ext{I}_0$, addressing open questions in the field.

Findings

Established strong consistency of $ ext{I}_ ext{omega}$

Embedded first-order intuitionistic logic into $ ext{I}_0$

Partially answered open questions about logical system embeddings

Abstract

We show a model construction for a system of higher-order illative combinatory logic $\mathcal{I}_\omega$, thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system $\mathcal{I}_0$ of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.