 To complete the first step we have to determine whether this result is robust. For it could be the case that when one considers the next level, Σ22 (or further levels, like third-order arithmetic) CH is no longer part of the picture, that is, perhaps large cardinals imply that there is an axiom A such that ZFC + A is Ω-complete for Σ22 (or, going further, all of third order arithmetic) and yet not all such A have an associated TA which contains CH. We must rule this out if we are to secure the first step. 