Understanding two lines from a proof that successor cardinals are regular

Question

I would appreciate help checking my thinking and understanding regarding the following two relations from a proof that: For every ordinal $\alpha$, the successor cardinal $\aleph_{\alpha}^{+}$ is regular.

If $S\subseteq\aleph_{\alpha}^{+}$ is cofinal ($\alpha$ is any ordinal)

1) $\aleph_{\alpha}^{+} = \bigcup_{\beta\in S}\beta$

I think that since $S$ is cofinal, then for every $x\in\aleph_{\alpha}^{+}$, there is a $y\in S$ such that $x\leq y$. Thus the union of the elements of $S$ is equal to $\aleph_{\alpha}^{+}$

2) $\aleph_{\alpha}^{+} \leq \sum_{\beta\in S} |\beta|$

I could see this if the summands were $\beta$'s; then the RHS would be an upper bound. Where I am confused is that for every ordinal $\beta$, the cardinality $|\beta|\leq \beta$. So how do you know 2) holds?

Thanks

Answer 1

1. You are almost correct. $S$ is cofinal, so - as you've noted - for every $\gamma \in \aleph_{\alpha}^{+}$ there is some $\sigma \in S$ such that $\gamma \leq \sigma$. Since $\aleph_{\alpha}^{+}$ is a limit ordinal we get a slighly better result (replacing $\gamma$ with $\gamma+1$): For $\gamma \in \aleph_{\alpha}^{+}$ there is some $\sigma \in S$ such that $\gamma +1 \leq \sigma$, i.e. $\gamma \in \sigma$. This now immediately implies (using the transitivity of ordinals) that $$ \bigcup S = \aleph_{\alpha}^{+}. $$

2. We need to find an injection $$ f \colon \aleph_{\alpha}^{+} \to \sum_{\sigma \in S} | \sigma |. $$ For each $\gamma \in \aleph_{\alpha}^{+}$ let $\sigma(\gamma)$ be the least $\sigma \in S$ such that $\gamma +1 \le \sigma$. (Such a $\sigma$ exists by our remark above). Furthermore fix, for each $\sigma \in S$, a bijection $$ \pi_{\sigma} \colon \sigma \to | \sigma |. $$ (Why is this possible?) Now recall that $\sum_{\sigma \in S} | \sigma | = \bigcup_{\sigma \in S} | \sigma | \times \{\sigma \}$ - the disjoint union of all $| \sigma |$. So, for each $\gamma \in \aleph_{\alpha}^{+}$, $f(\gamma)$ is of the form $$ f(\gamma) = (i, \sigma), $$ where $\sigma \in S$ is suitable and $i \in |\sigma| = \operatorname{ran}(\pi_{\sigma})$. (What is a suitable $\sigma$ and how could we choose $i \in | \sigma|$) to make $f$ into an injection? [Hint: Make use of $\sigma(\gamma)$ and $\pi_{\sigma(\gamma)}$.])