(a) Let $\kappa$ and $\lambda$ be cardinals. What does it mean to say that $\kappa<\lambda$ ? Explain briefly why, assuming the Axiom of Choice, every infinite cardinal is of the form $\aleph_{\alpha}$ for some ordinal $\alpha$, and that for every ordinal $\alpha$ we have $\aleph_{\alpha+1}<2^{2^{\aleph_{\alpha}}}$.

(b) Henceforth, you should not assume the Axiom of Choice.

Show that, for any set $x$, there is an injection from $x$ to its power set $\mathcal{P} x$, but there is no bijection from $x$ to $\mathcal{P} x$. Deduce that if $\kappa$ is a cardinal then $\kappa<2^{\kappa}$.

Let $x$ and $y$ be sets, and suppose that there exists a surjection $f: x \rightarrow y$. Show that there exists an injection $g: \mathcal{P} y \rightarrow \mathcal{P} x$.

Let $\alpha$ be an ordinal. Prove that $\aleph_{\alpha} \aleph_{\alpha}=\aleph_{\alpha}$.

By considering $\mathcal{P}\left(\omega_{\alpha} \times \omega_{\alpha}\right)$ as the set of relations on $\omega_{\alpha}$, or otherwise, show that there exists a surjection $f: \mathcal{P}\left(\omega_{\alpha} \times \omega_{\alpha}\right) \rightarrow \omega_{\alpha+1}$. Deduce that $\aleph_{\alpha+1}<2^{2^{\kappa_{\alpha}}}$.