The number of countable models in realizability toposes

The goal of this paper is to extend Morley’s results in Morley (1970) [9] to realizability toposes. We consider three natural notions of “countable model” in this context. We show for each of these notions of countable and for any first order theory T in a countable language, there is either a perfect set of non-isomorphic models of T or there are at most ℵ1 many non-isomorphic models of T in the realizability topos over any countable PCA.