Goodman’s theorem states that is conservative over . The same result applies to the extensional case, that is, is also conservative over . This is due to Beeson. In this article, we modify Goodman realizability and provide a new proof of the extensional case.
"On Goodman Realizability." Notre Dame J. Formal Logic 60 (3) 523 - 550, August 2019. https://doi.org/10.1215/00294527-2019-0018