Let $Y \supseteq^{*} X$ denote that all but finitely many members of X belong to Y. Recall that a tower is a linearly quasi-ordered subset of $(\mathcal{P}^{\omega}(\omega), \supseteq^{*})$ maximal with respect to upward extension (equivalently, with no infinite "almost intersections"; note that the notation $\supseteq^{*}$ corresponds here to reverse inclusion), and that the cardinal $\mathfrak{t}$ is the least cardinality of any tower. Van Douwen defines $\mathfrak{t}$ equivalently as the least cardinality of any well-ordered tower. It is not too hard to show that $\omega < \mathfrak{t} \leq \mathfrak{c}$ and that there exists a tower of cardinality $\mathfrak{c}$. However, it is less obvious to me that there exists a well-ordered tower of cardinality $\mathfrak{c}$.
In other words, does the poset of subsets of $\omega$ up to almost-equality, ordered with respect to almost-inclusion, admit a subset of order-type $\mathfrak{c}$?
My motivation for this question is that each of the cardinal characteristics of the continuum are usually defined as the minimum of sets of cardinals with trivial maximum, but it is not obvious to me whether the trivial maximum is attained in this otherwise natural alternative definition of $\mathfrak{t}$.
Best Answer
No, not necessarily. For instance, if you start with a model of CH and add any number of Cohen reals, there will never exist a well-ordered subset of $(\mathcal{P}^{\omega}(\omega), \supseteq^{*})$ of length $\omega_2$ (so in particular, if you add at least $\aleph_2$ Cohen reals, there will not exist a well-ordered tower of cardinality $\mathfrak{c}$). The idea of the proof is that given a set of more than $\aleph_1$ names for reals, we can find an automorphism of the forcing that swaps two of them (roughly because each name for a real depends on only countably many of our Cohen reals, but the forcing notion that adjoins countably many Cohen reals has only $\aleph_1$ different names for reals by CH). As a result, we cannot have a name for a set of more than $\aleph_1$ reals which is forced to be totally ordered by $\supseteq^*$ in a particular way (e.g., with order-type $\omega_2$), since that order would be violated by swapping any two of them.