Narrow Results

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