Generic absoluteness and the continuum

Let Hω2 denote the collection of all sets whose transitive closure has size at most א1. Thus, (Hω2 ,∈) is a natural model of ZFC minus the power-set axiom which correctly estimates many of the problems left open by the smaller and better understood structure (Hω1 ,∈) of hereditarily countable sets. One of such problems is, for example, the Continuum Hypothesis. It is largely for this reason that the structure (Hω2 ,∈) has recently received a considerable amount of study (see e.g. [15] and [16]). Recall the well-known Levy-Schoenfield absoluteness theorem ([10, §2]) which states that for every Σ0−sentence φ(x, a) with one free variable x and parameter a from Hω2 , if there is an x such that φ(x, a) holds then there is such an x in Hω2 , or in other words, (Hω2 ,∈) ≺1 (V,∈). (1)