The last forcing standing with diamonds

TL;DR

This paper introduces a new property of certain forcing notions that ensures their lambda-support iterations preserve the cardinality lambda^+ in set theory.

Contribution

It presents a novel property of (<lambda)-strategically complete forcing notions that guarantees the preservation of lambda^+ during iterations.

Findings

The new property implies non-collapse of lambda^+ in iterations.

It extends previous work by Roslanowski and Shelah on forcing notions.

Provides conditions for preserving cardinalities in forcing iterations.

Abstract

This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+.