Abstract Corrected 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 $({\mathbb{P}}_{\alpha },\underset{\sim }{{\mathbb{Q}}_{\beta }}:\alpha \le \delta ,\beta <\delta )$ 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 ${\mathbb{P}}_U⋖{\mathbb{P}}_{\delta }$.