Some Banach spaces added by a Cohen real

TL;DR

This paper investigates how adding a Cohen real to a model introduces new Banach spaces of density  that cannot embed into existing spaces, affecting universality numbers in set theory.

Contribution

It demonstrates that adding a Cohen real creates Banach spaces of density  with unique embedding properties, impacting the isomorphic universality number for such spaces.

Findings

Adding one Cohen real adds a -density Banach space not embeddable into ground model spaces.

The isomorphic universality number for Banach spaces of density  is  in the Cohen model.

Universality results extend to other cardinals with different proofs.

Abstract

We study certain Banach spaces that are added in the extension by one Cohen real. Specifically, we show that adding just one Cohen real to any model adds a Banach space of density $\aleph_1$ which does not embed into any such space in the ground model such a Banach space can be chosen to be UG This has consequences on the the isomorphic universality number for Banach spaces of density $\aleph_1$, which is hence equal to $\aleph_2$ in the standard Cohen model and the same is true for UG spaces. Analogous universality results for Banach spaces are true for other cardinals, by a different proof.