This is an announcement for the paper "Isomorphic universality and the number of pairwise non-isomorphic in the class of Banach spaces" by Mirna Dzamonja. Abstract: 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 universality of Banach spaces of density $\aleph_1$ (Theorem 1.5). 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)). Archive classification: math.LO math.FA Mathematics Subject Classification: 03E75, 46B26, 46B03, 03C45, 06E15 Submitted from: h020@uea.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.3640 or http://arXiv.org/abs/1308.3640