There exists a family $\{B_\alpha\}_{\alpha < \omega_1}$ of sets of countable ordinals such that: (1) $\max B_\alpha = \alpha$, (2) if $\alpha \in B_\beta$ then $B_\alpha \subseteq B_\beta$, (3) if $\lambda \leq \alpha$ and $\lambda$ is a limit ordinal then $B_\alpha \cap \lambda$ is not in the ideal generated by the $B_\beta, \beta < \alpha$, and by the bounded subsets of $\lambda$, (4) there is a partition $\{A_n\}^\infty_{n = 0}$ of $\omega_1$ such that for every $\alpha$ and every $n, B_\alpha \cap A_n$ is finite.