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 and for 0-dimensional and such that the following requirement holds for all and in : if , then there are constants and with and .
Mirna Džamonja. "Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces." Abstr. Appl. Anal. 2014 (SI26) 1 - 11, 2014. https://doi.org/10.1155/2014/184071