 The most optimistic scenario along these lines is this: The scenario is that there is a large cardinal axiom L and axioms A→ such that ZFC + L + A→ is Ω-complete for all of third-order arithmetic and all such theories are Ω-equivalent and imply CH. Going further, perhaps for each specifiable fragment Vλ of the universe of sets there is a large cardinal axiom L and axioms A→ such that ZFC + L + A→ is Ω-complete for the entire theory of Vλ and, moreover, that such theories are Ω-equivalent and imply CH. Were this to be the case it would mean that for each such λ there is a unique Ω-complete picture of Vλ and we would have a unique Ω-complete understanding of arbitrarily large fragments of the universe of sets. This would make for a strong case for new axioms completing the axioms of ZFC and large cardinal axioms. 