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I have read some of the related questions (cofinality and its consequences, for example) about cofinality, but I still have no idea how to calculate the cofinality of an aleph.

So, let's say I have a cardinal $\aleph_\alpha$, how should I go about calculating its cofinality?

I read some of Jech's Set Theory, but I did not grasp anything that might be useful to do my calculations. I apologize in advance if I missed something out from the book; I'll appreciate references to where I find the answer in the book.

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I think that [cofinality] is somewhat over-specific. This fits well into [cardinals] or [elementary-set-theory]. –  Asaf Karagila Jun 9 '11 at 10:57
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up vote 11 down vote accepted

Starting out with $\aleph_0$, since the union of finitely many finite sets is finite, it follows that $\text{cof}(\aleph_0)=\aleph_0$.

For successor cardinals, since $\aleph_{\alpha+1}$ means $(\aleph_\alpha)^+$, which is a regular cardinal (this uses AC), we know $\text{cof}(\aleph_{\alpha+1})=\aleph_{\alpha+1}$.

The remaining case is limit cardinals, or $\aleph_\lambda$ for a limit ordinal $\lambda$. In this case, $\aleph_\lambda=\text{sup}_{\alpha\lt\lambda}\aleph_\alpha$ is the limit of a $\lambda$-sequence of smaller cardinals, and so by passing to the shortest possible subsequence of this sequence, we see that $\text{cof}(\aleph_\lambda)=\text{cof}(\lambda)$. So for example, $\text{cof}(\aleph_{\aleph_3+\omega^3})=\omega$, since there is an $\omega$-sequence unbounded in the ordinal $\aleph_3+\omega^3$.

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