Collapsing successors of singulars

Abstract:

Let $\kappa$ be a singular cardinal in $V$, and let $W \supseteq V$ be a model such that $\kappa ^+_V = \lambda ^+_W$ for some $W$-cardinal $\lambda$ with $W \models \operatorname {cf}(\kappa ) \neq \operatorname {cf}(\lambda )$. We apply Shelah’s pcf theory to study this situation, and prove the following results.

1. $W$ is not a $\kappa ^+$-c.c generic extension of $V$.

2. There is no “good scale for $\kappa$” in $V$, so in particular weak forms of square must fail at $\kappa$.

3. If $V \models \operatorname {cf}(\kappa ) = \aleph _0$ then $V \models {}$ “$\kappa$ is strong limit $\implies 2^\kappa = \kappa ^+$”, and also ${}^\omega \kappa \cap W \supsetneq {}^\omega \kappa \cap V$.

4. If $\kappa = \aleph _\omega ^V$ then $\lambda \le (2^{\aleph _0})_V$.