Is it possible to find a sequence $(A_\alpha)_{\alpha < \omega_1}$ in $\mathcal P(\omega)$ with the following property:
(1) If $\alpha < \beta$, then $|A_\alpha \setminus A_\beta|$ is finite but $|A_\beta \setminus A_\alpha|$ is infinite;
$\mathcal P(\omega)$ is the powerset of $\omega$. I'm really at a loss as to how to approach this, so any hints are appreciated. For what it's worth, an affirmative answer to this question was assumed in an answer to another question of mine at MO.
I would recursively construct sets $B_\xi$ for $\xi<\omega_1$ so that $B_\eta\subset^*B_\xi$ whenever $\xi<\eta<\omega_1$ and then let $A_\xi=\omega\setminus B_\xi$ for each $\xi<\omega_1$. (Here $X\subset^*Y$ means that $|X\setminus Y|<\omega=|Y\setminus X|$.)
Suppose that $\eta<\omega_1$, and we have infinite subsets $B_\xi$ of $\omega$ for $\xi<\eta$ such that $B_\xi\subset^*B_\zeta$ whenever $\zeta<\xi<\eta$. Enumerate $\{B_\xi:\xi<\eta\}$ as $\{C_k:k\in\omega\}$, recursively choose distinct $m_k,n_k\in\omega$ so that $m_k,n_k\in\bigcap_{i\le k}C_i$ for each $k\in\omega$, let $B_\eta=\{n_k:k\in\omega\}$, and check that $B_\eta\subset^*B_\xi$ for each $\xi<\eta$.