In the book "Sets, Logic and Categories" by Peter Cameron it is stated that, if $\lambda$ is a limit ordinal then $\aleph _\lambda =\bigcup _{\beta< \lambda} \aleph _\beta$ is a cardinal.
The proof from the book:
"It is not obvious that $\aleph_\lambda$ is a cardinal. It is certainly an ordinal, since it is a union of ordinals.
Suppose that it were bijective with a section of itself. This section could not contain all $\aleph_\beta$; but if some $\aleph_\beta$ does not lie in the section, then the restriction of the bijection takes $\aleph_\beta$ into a section of itself, a contradiction."
My understanding:
The author wants to show that there is no bijection between $\aleph_\lambda$ and any of its sections for it to be a cardinal. But I have trouble understanding the last sentence:
What does "the restriction of the bijection takes $\aleph_\beta$ into a section of itself" mean?
It’s not immediately clear that the restriction of the bijection takes $\aleph_\beta$ into a section of itself rather than simply into a proper subset of itself, but the argument can be repaired. If there is an $\aleph_\gamma$ that is not a cardinal, there is a least such, and it clearly cannot be a successor cardinal, so it must be $\aleph_\gamma=\bigcup_{\beta<\lambda}\aleph_\beta$ for some limit ordinal $\lambda$.
Suppose that $S$ is a section of $\aleph_\gamma$, and $f:\aleph_\gamma\to S$ is a bijection. There is a $\beta<\lambda$ such that $\aleph_\beta\notin S$, so $S\subseteq\aleph_\beta$. But then $f\upharpoonright\aleph_{\beta+1}$ is an injection from $\aleph_{\beta+1}$ to $\aleph_\beta$, which is impossible.