Definition of cofinality

Question

Let $\alpha$ be a limit ordinal. We define $\operatorname{cf}\alpha$ to be the least limit ordinal $\beta$ such that there is an increasing $\beta$-sequence $\langle \alpha_{\xi} \mid \xi < \beta\rangle$ with $\displaystyle\lim_{\xi \to \beta} \alpha_{\xi} = \alpha$ (This is how Jech defines it).

But how do I define cofinality of a non-limit ordinal? For example $\operatorname{cf}(4)$, or $\operatorname{cf}(\omega + 5)$? This comes up in my reading frequently and I'm not sure how to deal with it.

Thanks very much.

Answer 1

The cofinality of a partially ordered set $P$ is defined to be the least cardinality of all cofinal sets in $P$.

A set $B \subset A$ is called cofinal in $A$ if for every $a \in A$ there is a $b \in B$ such that $a \leq b$.

Now applying this to $4$ we observe that $\{ 3 \}$ is cofinal in $4$ and hence $\operatorname{cf}{(4)} = |\{ 3 \}| = 1$.

Similarly, we observe that the set $\{ \omega + 4 \}$ is cofinal in $\omega + 5$, hence $\operatorname{cf}{(\omega + 5)} = 1$.

Note that every successor ordinal has an $\in$-maximal element and hence has cofinality $1$.

Hope this helps.

Answer 2

The general definition of cofinality would be $0$ if $\alpha=0$; $1$ if $\alpha=\beta+1$; and the limit case you already know.

This is really extending the limit case definition since what we really care about is "how long" is the shortest set which is unbounded (note that cofinal and unbounded are the same thing here since this is a linear order). If $\alpha=\beta+1$ then $\{\beta\}$ is unbounded in $\alpha$ and has order type $1$.

The thing is that we really don't care about cofinality of successor ordinals. Note that if $\kappa$ has an uncountable cofinality then the set of successor ordinals below $\kappa$ is non-stationary, so it actually has very little play in telling us what happens at the end.

Furthermore, since the intersection of stationary and club is stationary, you can always intersect your stationary set with the club of limit ordinals, thus assume that there are no successor ordinals in your considerations to begin with.

Answer 3

Note that if $\alpha$ is not a limit ordinal then there are uniquely determined ordinals $\lambda$ and $n$ such that $\alpha=\lambda+n$, where $\lambda$ is a limit and $n<\omega$. Thus, you may define the cofinality of $\alpha$ to be 1.

This procedure is, however, redundant, as the definition of cofinality may be easily extended to subsets of partially ordered sets.