Exercises for Week 5

1. Prove that the following statements are equivalent, modulo the other axioms that we have seen.
   1. The Axiom of Choice: every set of nonempty sets has a choice function.
   2. The product of a (set-sized) family of nonempty sets is nonempty.
   3. Every (set-sized) binary relation has a uniformization: It contains the graph of a function with the same domain.
2. Using the Axiom of Choice, prove that for every set $A$ there is a well-ordering $R$ of $A$.
3. Prove or disprove the following statement: There is a cofinal, (strictly) increasing function $\aleph_1 \to \aleph_\omega$.
4. For a family of sets $(X_i : i \in I)$ we define the sum (or "disjoint union") $\sum_{i \in I} X_i = \{(i,x) : i \in I \text{ and } x \in X_i\}$. Prove that the following statements are equivalent for any infinite cardinal $\kappa$.
   1. $\kappa$ is singular.
   2. $\kappa = \bigcup_{i \in I} X_i$ for some family of sets $(X_i : i \in I)$ with $|I| \lt \kappa$ and each $|X_i| \lt \kappa$.
   3. $\kappa = \left| \sum_{i \in I} X_i \right|$ for some family of sets $(X_i : i \in I)$ with $|I| \lt \kappa$ and each $|X_i| \lt \kappa$.
5. For an ordinal $\kappa$ and a set $X$ we define $X^{\mathord{\lt}\kappa} =\bigcup_{\alpha \lt \kappa} X^\alpha$. If $\kappa$ is an infinite cardinal we say that $\kappa$ is a strong limit cardinal if $\forall \alpha \lt \kappa\,(|2^\alpha| \lt \kappa)$. (Note that any strong limit cardinal is a limit cardinal by Cantor's theorem.) Prove that if $\kappa$ is a strong limit cardinal then $\left|\kappa^{\text{cf}(\kappa)}\right| = |2^\kappa|$.