Exercises for Week 4

1. The aleph sequence $(\aleph_\alpha : \alpha \in \operatorname{Ord})$ is defined by transfinite recursion: $\aleph_0 = \omega$, $\aleph_{\alpha + 1} = (\aleph_\alpha)^+$, and $\aleph_\lambda = \sup_{\alpha \lt \lambda} \aleph_\alpha$ if $\lambda$ is a limit ordinal. Prove that an ordinal $\kappa$ is a cardinal if and only if $\kappa = \aleph_\alpha$ for some ordinal $\alpha$.
2. Let $X$, $Y$, and $Z$ be sets. Prove the following statements.
   1. $Z^{X\cup Y} \approx Z^X \times Z^Y$ if $X \cap Y = \emptyset$.
   2. $Z^{Y \times X} \approx (Z^Y)^X$.
   3. $2^X \approx \mathcal{P}(X)$.
   4. $X^2 \approx X \times X$.
3. Let $\kappa$ and $\lambda$ be cardinals. Define a relation $\lt_\text{lex}$ on $\kappa^\lambda$ by $f \lt_\text{lex} g$ if $f \ne g$ and $f(\alpha) \lt g(\alpha)$ for the least ordinal $\alpha \lt \lambda$ with $f(\alpha) \ne g(\alpha)$. Prove the following statements.
   1. $\kappa^\lambda$ is linearly ordered by $\lt_\text{lex}$.
   2. $\kappa^\lambda$ is not well-ordered by $\lt_\text{lex}$ if $\lambda$ is infinite and $\kappa \ge 2$.
4. Prove that if $\lambda$ is a limit ordinal, then $\text{cf}(\aleph_\lambda) = \text{cf}(\lambda)$.
5. Prove that if $\lambda$ is a limit ordinal, then $\text{cf}(\lambda)$ is the least ordinal $\alpha$ such that there is a cofinal function $\alpha \to \lambda$. (If we replaced "cofinal function" with "cofinal increasing function" then this would say something we already know.)