 The two situations are parallel with regard to maximality but in terms of the level of Ω-completeness the first is stronger. For in the first case we are not just getting Ω-completeness with regard to the Π2 theory of H(ω2) (with the additional predicates), rather we are getting Ω-completeness with regard to all of H(ω2). This is arguably an argument in favour of the case for ¬CH, even granting the conjecture. 