Equivalent condition for a cardinal number to be of cofinality $\aleph_0$

Question

Let cf$(\alpha)$ denotes the co-final of the transfinite cardinal number $\alpha.$ For every successor cardinal $\alpha$ we have cf$(\alpha)=\alpha\neq \aleph_0.$ Thus cf$(\alpha) = \aleph_0,$ implies $\alpha$ is a limit cardinal.

We can find some limit cardinal number $\alpha,$ for example $\alpha=\aleph_{\omega}$ with cf$(\alpha)=\aleph_0.$ Now, can we fined all (limit) cardinal numbers which satisfies cf$(\alpha)=\aleph_0?$

Answer 1

Asaf Karagila has essentially answered this question in comments (and if he posts an answer I'll delete this one for precedence reasons), but let me flesh it out a bit.

Remember that cofinality makes sense for arbitrary infinite limit ordinals, not just infinite cardinals. The characterization is:

$\aleph_\alpha$ has cofinality $\omega$ iff $\alpha=0$ or $\alpha$ (is a limit ordinal and) has cofinality $\omega$.

As an aside, remember that $\aleph_\alpha$ and $\omega_\alpha$ are literally the same thing, the difference between the notation is merely that the former indicates that we're thinking about only cardinals while the second indicates that we're thinking about general ordinals. I'm going to use both notations since in this particular question I think it will be clearer, but really I think the ideal approach is to ditch the $\aleph$-notation completely except for situations when we're using cardinal arithmetic - that is cardinal addition, multiplication, and exponentiation (which we're not doing here).

So, for example, the cardinals $\aleph_\omega, \aleph_{\omega^3+\omega^2\cdot 13}$, and $\aleph_{\omega_{\omega_\omega}}$ each have cofinality $\omega$, while $\aleph_{37}$ and $\aleph_{\omega_1}$ do not since $cf(37)=1$ and $cf(\omega_1)=\omega_1$.

Proving the fact above is a good exercise; here are a couple hints:

• In the right-to-left direction, suppose $cf(\alpha)=\omega$. Let $(\beta_i)_{i\in\omega}$ be cofinal in $\alpha$; what can you say about the sequence $(\aleph_{\beta_i})_{i\in\omega}$?

• In the left-to-right direction, we do the same thing but in the other direction. Suppose $\aleph_\alpha$ has cofinality $\omega$ and $\alpha\not=0$. Let $(\aleph_{\beta_i})_{i\in\omega}$ be cofinal in $\aleph_\alpha$: what can you say about the sequence $(\beta_i)_{i\in\omega}$, and what does that tell us about $\alpha$?