On Goodman Realizability

Abstract

Goodman’s theorem states that ${\mathsf{HA}}^{\omega }+\mathsf{AC}+\mathsf{RDC}$ is conservative over $\mathsf{HA}$. The same result applies to the extensional case, that is, $\mathsf{E}\text{-}{\mathsf{HA}}^{\omega }+\mathsf{AC}+\mathsf{RDC}$ is also conservative over $\mathsf{HA}$. This is due to Beeson. In this article, we modify Goodman realizability and provide a new proof of the extensional case.