Preservation properties for iterations with finite support

TL;DR

This paper studies how certain unbounded families related to classical cardinal invariants are preserved during ccc forcing iterations with finite support, extending results to direct limits and discussing ultrapower forcing with measurable cardinals.

Contribution

It extends preservation results of unbounded families in ccc iterations to direct limits and provides an overview of Shelah's ultrapower forcing theory with measurable cardinals.

Findings

Preservation of unbounded families in ccc iterations with finite support.

Extension of preservation properties to direct limits of iterations.

Overview of Shelah's ultrapower forcing with measurable cardinals.

Abstract

We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some classical cardinal invariant. We also prove that, under some conditions, these preservation properties can be preserved in direct limits of an iteration, so applications are extended beyond the context of finite support iterations. Also, we make a breve exposition of Shelah's theory of forcing with an ultrapower of a poset by a measurable cardinal.