Adding a lot of Cohen reals by adding a few I

TL;DR

This paper constructs models of set theory where adding a small number of Cohen reals to a larger model results in a significantly larger number of Cohen reals in a smaller submodel, exploring the interplay between models and Cohen forcing.

Contribution

It introduces a method to produce models where adding Cohen reals to a larger model yields more Cohen reals in a submodel, highlighting new interactions in set-theoretic forcing.

Findings

Adding κ Cohen reals to V₂ adds λ Cohen reals to V₁, with λ > κ

Models where V₁ and V₂ share the same cardinals are constructed

Demonstrates control over Cohen real addition across models

Abstract

In this paper we produce models $V_1\subseteq V_2$ of set theory such that adding $\kappa$-many Cohen reals to $V_2$ adds $\lambda$-many Cohen reals to $V_1$, for some $\lambda>\kappa$. We deal mainly with the case when $V_1$ and $V_2$ have the same cardinals.