Let $(P, \le)$ be a partially ordered set. A subset $A\subseteq P$ is a chain if any two elements of its elements are comparable. A subset $A\subseteq P$ is cofinal if every element of $P$ is less than or equal to some element of $A$.
Not every partially ordered set admits a cofinal chain (for example the poset of all finite subsets of an uncountable set, ordered by inclusion, has no cofinal chain). For those posets that do, there are in general many such chains.
Every totally ordered set $Q$ (in particular every chain in $P$) admits a cofinal well-ordered subset, and the smallest cardinality of such a subset is called the cofinality of $Q$.
Now the question:
Let $P$ be a partially ordered set with a cofinal chain. Do all the cofinal chains in $P$ have the same cofinality?
Proof or counterexample?
(If $P$ has a maximum element, all cofinal chains go through that element and have cofinality $1$, the trivial case. So we can assume $P$ has no maximum element.)
Best Answer
I think they do, by a similar argument to this fact for total orders. Suppose $P$ has no maximum element and we have regular ordinals $\alpha$ and $\beta$ and cofinal embeddings $i: \alpha \to P$ and $j: \beta \to P$ (we may assume $\alpha$ and $\beta$ are regular by passing to a cofinal subset of minimal order type).
Then for every $\gamma < \alpha$, we can pick some $\delta_\gamma < \beta$ with $j(\delta_\gamma) > i(\gamma)$, by cofinality of $j(\beta)$. The set $\{\delta_\gamma \mid \gamma < \alpha\}$ is cofinal in $\beta$, since for any $\delta_0 < \beta$, we can choose some $\gamma_0 < \alpha$ with $i(\gamma_0) > j(\delta_0)$, by cofinality of $i(\alpha)$. Then $\delta_{\gamma_0} > \delta_0$, because $j(\delta_{\gamma_0}) > i(\gamma_0) > j(\delta_0)$.
Hence $\mathrm{cf}(\beta) \le \lvert \alpha \rvert = \alpha = \mathrm{cf}(\alpha)$.
By symmetry we also have $\mathrm{cf}(\alpha) \le \mathrm{cf}(\beta)$. So all cofinal chains have the same cofinality.
There are many ways to present this argument of course. It's a bit simpler if your definition of cofinality is just "smallest cardinality of a cofinal set". You could also deduce it directly from this property of total orders by extending the ordering on $P$ to a total order, which is a standard Zorn's lemma argument.