Cardinal collapsing and product forcing

TL;DR

This paper investigates how certain product forcings can collapse the continuum at a singular strong limit cardinal of countable cofinality, providing new insights into cardinal arithmetic and forcing techniques.

Contribution

It demonstrates that forcing with a full product of adding Cohen subsets collapses the continuum at a singular strong limit cardinal, answering a question of Sy Friedman.

Findings

Forcing with the product collapses 2^κ into κ^+ under certain conditions.

Provides a new proof relating product forcing to adding a Cohen subset at κ^+.

Shows the product forcing is equivalent to adding a Cohen subset at κ^+.

Abstract

Suppose $\kappa$ is a singular strong limit cardinal of countable cofinality and let $\langle \kappa_{n}: n<\omega \rangle$ 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 \[ \prod_{n<\omega}Add(\kappa_n,1)/fin \simeq Add(\kappa^+, 1). \]