$Θ$-Hilbertianity and strong $Θ$-Hilbertianity

Abstract

Θ-Hilbertianity and its strengthening, strong Θ-Hilbertianity, are two generalizations of Hilbertianity inspired by Jarden's definition of p-Hilbertianity and strong p-Hilbertianity. Jarden has asked whether the two notions defined by him are actually the same. We address this question in its more general version of Θ-Hilbertianity and show that for PRC, and, in particular, for PAC fields, p-Hilbertianity and strong p-Hilbertianity coincide.