TITLE: What is the cofinality of $\omega+1$?
QUESTION [6 upvotes]: so $\omega+1=\{\{0,1,2,...\},\{\omega\}\}$, it seems to be the case that $cf(\omega+1)=1$, since $1=|\{\omega\}|$ is the least cardinality of cofinal subsets of $\omega+1$, but $1$ is also the cofinality of the ordinal $2$, which is much less than $\omega+1$. How can I overcome this contradiction?

REPLY [9 votes]: There's no contradiction; cofinality is not a strictly increasing function. Indeed, the cofinality of every non-zero successor ordinal (i.e., ordinals $\alpha$ such that $\alpha = \beta + 1$ for another ordinal $\beta$) is $1$.
One way to think about cofinality of an ordinal $\alpha$ is that it measures how long a sequence needs to be in order to "reach" $\alpha$, not how big $\alpha$ is.