  Realizability notions in mathematical logic have a long history, which can be
traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the
foundations of intuitionistic logic. Kleene's initial realizability laid the
ground for more sophisticated notions such as Kreisel's modified realizability
and various modern approaches. In this context, our work aligns with the
lineage of realizability strategies that emphasize the accumulation, rather
than the propagation of precise witnesses. In this paper, we introduce 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, which enables the gathering of witnesses into nonempty finite sets.
We also show that the previous analysis can be extended from logic to (Heyting)
arithmetic.
