Cohen Preservation and Independence

TL;DR

This paper introduces a general preservation theorem for independent families in set theory, demonstrating how certain forcing iterations maintain independence properties and applying it to specific models like the Miller Lite model.

Contribution

It provides a new, broad preservation theorem for independent families under countable support iterations, extending previous results and introducing Cohen preservation.

Findings

Theorem applies to various forcing iterations preserving independence.

Shows $rak{i} = eth_1$ in the Miller Lite model.

Establishes Cohen preservation in the context of independent families.

Abstract

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $\mathfrak{i}$ is strictly below $\mathfrak{c}$, including iterations of Sacks forcing, Miller partition forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of "Cohen preservation", which we discuss in detail.