Abstract Corrected Iterations

TL;DR

This paper extends the theory of forcing iterations by demonstrating that certain corrected $(< ext{-support})$-support iterations of definable forcing notions preserve elementary substructure relations, generalizing a known result for Suslin ccc forcing.

Contribution

It introduces a method to correct $(< ext{-support})$-iterations of definable forcing notions, establishing a new preservation property analogous to Judah and Shelah's result for Suslin ccc forcing.

Findings

Corrected iterations preserve elementary substructure relations.

Generalizes Judah and Shelah's result to broader classes of forcing.

Provides a framework for analyzing complex forcing iterations.

Abstract

We consider $(<\lambda)$-support iterations of a version of $(<\lambda)$-strategically complete $\lambda^+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by Judah and Shelah for finite support iterations of Suslin ccc forcing, namely that if $(\mathbb{P}_{\alpha}, \underset{\sim}{\mathbb Q_{\beta}} : \alpha \leq \delta, \beta <\delta)$ is a FS iteration of Suslin ccc forcing and $U\subseteq \delta$ is sufficiently closed, then letting $\mathbb{P}_U$ be the iteration along $U$, we have $\mathbb{P}_U \lessdot \mathbb{P}_{\delta}$.