Exercises for Week 3

Due Tuesday, January 29
  1. Let $x$ be a transitive set. Must its power set $\mathcal{P}(x)$ also be transitive? (Prove or disprove.)
  2. Assume the following lemmas. (1) For ordinals $\alpha$ and $\beta$ we have $\alpha \subseteq \beta$ if and only if $\alpha \in \beta$ or $\alpha = \beta$. (2) For ordinals $\alpha$ and $\beta$, the intersection $\alpha \cap \beta$ is also an ordinal. (3) If $\alpha$ is an ordinal then $\alpha \notin \alpha$. Prove that the membership relation $\in$ satisfies trichotomy on $\text{Ord}$.
  3. Prove that if $X$ is a set of ordinals, then $\bigcup X$ is an ordinal. (You may use the result from class that the relation $\in$ is a well-ordering of $\text{Ord}$, even though it depended on Exercise 2.)