Tower number is a regular uncountable cardinal

Question

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).

Answer 1

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