Herbrandized modified realizability

Realizability notions in mathematical logic have a long history, which can be tracedback to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations ofintuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticatednotions such as Kreisel’s modified realizability and various modern approaches. Inthis context, our work aligns with the lineage of realizability strategies that emphasizethe accumulation, rather than the propagation of precise witnesses. In this paper, weintroduce a new notion of realizability, namely herbrandized modified realizability.This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, whichenables the gathering of witnesses into nonempty finite sets. We also show that theprevious analysis can be extended from logic to (Heyting) arithmetic. ; The authors are grateful to Fernando Ferreira for interesting discussions on the topic. They extend their gratitude to the anonymous referee for providing valuable suggestions, which inspired the addition of Sect. 4.3 to the manuscript. Both authors acknowledge the support of Fundação para a Ciência e a Tecnologia under the Projects: UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020 and are also grateful to Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (Universidade de Lisboa). The first author is also grateful to LASIGE - Computer Science and Engineering Research Centre Herbrandized modified realizability (Universidade de Lisboa). The second author also benefitted from Fundação para a Ciência e a Tecnologia doctoral Grant 2022.12585.BD. ; info:eu-repo/semantics/publishedVersion