Tower number is a regular uncountable cardinal Let $A$, $B$ be subsets of $\omega$. We write $A \subset^* B$ when $A \setminus B$ is finite. A sequence of distinct infinite subsets of $\omega$ is called a tower if $A_\beta \subset^* A_\alpha$ whenever $\alpha< \beta$. The {\bf tower number} is the minimal length of a maximal tower (a tower such that no further set is almost contained in every member of that tower).

If $\alpha_n$ is a cofinal sequence, the collection of $A_{\alpha_n}$ is also a tower.

If $\\{A_n: n \in \omega\\}$ is a countable tower, we can take $b_1\in A_1$, $b_2\in A_1\cap A_2$,etc. $B = \\{b_n\\}$ is then almost contained in every $A_n$, so the tower is not maximal