Adding a lot of Cohen reals by adding a few II

TL;DR

This paper investigates how adding Cohen reals over an extension model can significantly increase the number of Cohen reals over the original model, revealing intricate relationships between models of set theory.

Contribution

It introduces a framework for understanding how adding Cohen reals over an extension model can produce many more reals over the base model, highlighting new interactions in set-theoretic forcing.

Findings

Adding Cohen reals over an extension can drastically increase the number of reals over the original model.

The paper characterizes conditions under which this increase occurs.

It provides new insights into the structure of models of ZFC under Cohen forcing.

Abstract

We study pairs $(V, V_{1})$, $V \subseteq V_1$, of models of $ZFC$ such that adding $\kappa-$many Cohen reals over $V_{1}$ adds $\lambda-$many Cohen reals over $V$ for some $\lambda> \kappa$.