Iterability for (transfinite) stacks

Abstract

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