Iterability for (transfinite) stacks

Abstract

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $\Omega$ be a regular uncountable cardinal. Let $m<\omega$ and $M$ be an $m$-sound premouse and $\Sigma$ be an $(m,\Omega+1)$-iteration strategy for $M$ (roughly, a normal $(\Omega+1)$-strategy). We define a natural condensation property for iteration strategies, "inflation condensation". We show that if $\Sigma$ has inflation condensation then $M$ is $(m,\Omega,\Omega+1)^*$-iterable (roughly, $M$ is iterable for length $\leq\Omega$ stacks of normal trees each of length ${<\Omega}$), and moreover, we define a specific such strategy $\Sigma^{\mathrm{st}}$ and a reduction of stacks via $\Sigma^{\mathrm{st}}$ to normal trees via $\Sigma$. If $\Sigma$ has the Dodd-Jensen property and $\mathrm{card}(M)<\Omega$ then $\Sigma$ has inflation condensation. We also apply some of the techniques developed to prove that if $\Sigma$ has strong hull condensation (introduced independently by John Steel) and $G$ is $V$-generic for an $\Omega$-cc forcing, then $\Sigma$ extends to an $(m,\Omega+1)$-strategy $\Sigma^+$ for $M$ with strong hull condensation, in the sense of $V[G]$. Moreover, this extension is unique. We deduce that if $G$ is $V$-generic for a ccc forcing then $V$ and $V[G]$ have the same $\omega$-sound, $(\omega,\Omega+1)$-iterable premice which project to $\omega$.