= Rank cutoff at the first inaccessible limit of inaccessibles
{title2=$V_\iota\models\mathrm{ZFC}+\infty\mathrm{IC}+\neg\mathrm{ILI}$}
If $\iota$ is the least <inaccessible limit of inaccessibles>, $V_\iota$ models <ZFC> and internally has unboundedly many <inaccessible cardinals>, while none of its ordinals is an <inaccessible limit of inaccessibles>. Rank agreement makes inaccessibility below $\iota$ absolute. If a model of unboundedly many inaccessibles has no such limit in the first place, no cutoff is needed. This proves the corresponding relative-consistency nonimplication.
Back to article page
