Isomorphic universality and the number of pairwise non-isomorphic models in the class of Banach spaces

We study isomorphic universality of Banach spaces of a given density and a number of pairwise non-isomorphic models in the same class. We show that in the Cohen model the isomorphic universality number for Banach spaces of density $\aleph_1$ is $\aleph_2$, and analogous results are true for other cardinals (Theorem 1.2(1)) and that adding just one Cohen real to any model destroys the old universality of Banach spaces of density $\aleph_1$ (Theorem 1.5). Moreover, adding one Cohen real adds a weakly compactly generated Banach space of density $\aleph_1$ which does not embed into any Banach space of density $\aleph_1$ in the ground model. We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces and 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 embeddings (very positive embeddings), is high (Theorem 4.8(1)), and similarly for the number of pairwise non-isomorphic models (Theorem 4.8(2)).