Define the cardinals $\aleph_{\alpha}$, and explain briefly why every infinite set has cardinality $\operatorname{an} \aleph_{\alpha} .$

Show that if $\kappa$ is an infinite cardinal then $\kappa^{2}=\kappa$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be infinite sets. Show that $X_{1} \cup X_{2} \cup \cdots \cup X_{n}$ must have the same cardinality as $X_{i}$ for some $i$.

Let $X_{1}, X_{2}, \ldots$ be infinite sets, no two of the same cardinality. Is it possible that $X_{1} \cup X_{2} \cup \ldots$ has the same cardinality as some $X_{i}$ ? Justify your answer.