Cardinal collapsing and product forcing

Abstract

Suppose $\kappa$ is a singular strong limit cardinal of countable cofinality and let $⟨{\kappa }_n:n<\omega ⟩$ be an incrasing sequence of regular cardinals cofinal in $\kappa$. We show that if $cf(2^{\kappa })={\kappa }^{+}$, then forcing with the full product ${\prod }_{n<\omega }Add({\kappa }_n,1)$ collapses $2^{\kappa }$ into ${\kappa }^{+}$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length ${\kappa }^{+},$ then forcing with ${\prod }_{n<\omega }Add({\kappa }_n,1)$ adds a generic filter for $Add({\kappa }^{+},1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)/fin \simeq Add(κ^+, 1). \]