Logic Seminar

We describe the conversion of a ladder system on $\omega_n$ to a nonzero element of $\mathrm{H}^n(\omega_n)$ (computed with respect to a constant sheaf on $\omega_n$). In the case of $n=1$, this process of conversion is, simply, Todorcevic's technique of walks on the countable ordinals -- one which, ``[d]espite its simplicity [...] can be used to derive virtually all known other structures that have been defined so far on $\omega_1$'' (\textit{Walks on Ordinals}, 19). For higher $n$, this conversion plays a more prospective role, pointing to dramatically underexplored ZFC structures on higher $\omega_n$. Time permitting, we'll outline two questions in this area that we think of as \textit{next}.