Corrected Iteration

TL;DR

This paper introduces a corrected iteration method for certain large cardinal forcing notions to preserve properties during iteration, enabling new consistency results in set theory.

Contribution

It proposes a novel correction technique for $(< ext{lambda})$-support iterations of $(< ext{lambda})$-complete forcing notions, improving preservation properties.

Findings

Successfully restores preservation in iterations of specific forcing notions.

Enables consistency results related to covering and dominating numbers.

Provides a framework for future iterations with similar properties.

Abstract

For $\lambda$ inaccessible, we may consider $(< \lambda)$-support iteration of some specific $(<\lambda)$-complete $\lambda^+$-c.c. forcing notion. But this fails a "preservation by restricting to a sub-sequence of the forcing, we "correct" the iteration to regain it. This is used in another paper in the consistency of $cov(meagre) < \mathfrak{d}_\lambda$.