Cohen Forcing and Inner Models

TL;DR

This paper investigates the relationship between Cohen forcing notions in an inner model and the ambient universe, establishing conditions under which they are equivalent or differ in strength, with generalizations to larger cardinals and iterations.

Contribution

It provides new criteria for comparing Cohen forcing from inner models and the universe, extending to multiple stages and arbitrary inner models.

Findings

Forcing from an inner model is at least as strong as from the universe iff the posets have the same cardinality.

A sufficient condition is given for the inner model forcing to be weaker than the universe forcing.

Results are generalized to $Add(oldsymbol{ ext{k}},oldsymbol{ ext{ extlambda}})$ and iterative Cohen forcing.

Abstract

Given an inner model $W \subset V$ and a regular cardinal $\kappa$, we consider two alternatives for adding a subset to $\kappa$ by forcing: the Cohen poset $Add(\kappa,1)$, and the Cohen poset of the inner model $Add(\kappa,1)^W$. The forcing from $W$ will be at least as strong as the forcing from $V$ (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from $V$ to fail to be as strong as that from $W$. The results are generalized to $Add(\kappa,\lambda)$, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.