Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces

We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding between C(K) and C(L) for 0‐dimensional K and L such that the following requirement holds for all h ≠ 0 and f ≥ 0 in C(K): if 0 ≤ Th ≤ Tf, then there are constants a ≠ 0 and b with 0 ≤ a · h + b ≤ f and a · h + b ≠ 0.