Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters](https://en.wikipedia.org/wiki/Ultrafilter) on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq\_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$


It is easy to see that $\leq\_{RK}$ is reflexive and transitive, but not anti-symmetric. Set ${\cal U}\simeq\_{RK} {\cal V}$ if ${\cal U}\leq\_{RK}{\cal V}$ and ${\cal V}\leq\_{RK}{\cal U}$. So $\text{NPU}(\omega)/\simeq\_{RK}$ is a poset with the Rudin-Keisler order applied to equivalence classes.


**Question.** If ${\cal C},{\cal D}$ are totally ordered subsets of $\text{NPU}(\omega)/\simeq\_{RK}$ that are maximal with respect to $\subseteq$, do they necessarily have the same cardinality?



Cardinality of maximal chains in the poset of ultrafilters with Rudin-Keisler ordering
Yes. I claim that such a maximal chain must always have cardinality $\frak{c}^{+}$. 


First of all, take notice that for each ultrafilter $U$, there are at most $\frak{c}$ many Rudin-Kiesler equivalence classes below $U$ simply because there are at most continuumly many functions $f:\omega\rightarrow\omega$ and every ultrafilter RK-below $U$ is of the form $\{R\subseteq\omega|f^{-1}[R]\in U\}$ for some function $f$.


Now observe that for every collection $\mathcal{C}$ of at most continuumly many ultrafilters, by [my answer here](https://mathoverflow.net/questions/131062/importance-of-separability-vs-second-countability/131122#131122) there is an ultraftiler $U$ with $V\leq\_{RK}U$ for each $V\in\mathcal{C}$.


Suppose now that $\mathcal{C}$ is a maximal chain in the collection of ultrafilters. Now, let $(U\_{\alpha})\_{\alpha<\lambda}$ be a strictly increasing cofinal sequence where $\lambda$ is some regular cardinal. Then since every collection of at most continuumly many ultrafilters on $\omega$ has a RK-upper bound, we conclude that $\lambda>\frak{c}$. On the other hand, since each $U\_{\alpha}$ has at most continuumly many $RK$-smaller ultrafilters, we conclude that $\lambda=\frak{c}^{+}$. Since each $U\_{\alpha}$ has at most continuumly many predecessors, we coonclude that $|\mathcal{C}|=\mathfrak{c}^{+}$ as well.


