Exercises for Week 4

1. Using our definition of "$+$" in terms of $<_{\text{lex}}$,
   1. Prove or disprove: $S(\alpha + \beta) = S(\alpha) + \beta$ holds for all ordinals $\alpha$ and $\beta$.
   2. Prove that $S(\alpha + \beta) = \alpha + S(\beta)$ holds for all ordinals $\alpha$ and $\beta$.
   Hint: By the definition of addition we have an isomorphism $F$ of the structure $(\alpha+\beta;\in)$ with the structure $(\{0\} \times \alpha \cup \{1\} \times \beta; <_{\text{lex}})$. Try to extend $F$ to get an isomorphism $G$ of the structure $(S(\alpha+\beta); \in)$ with something.
2. 1. Using the ordinal identities $\alpha \cdot S(\beta) = \alpha\cdot \beta + \alpha$ and $S(\alpha + \beta) = \alpha + S(\beta)$ together with the continuity of "$+$" and "$\cdot$" in the second variable, prove that \[\alpha\cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma\] for all ordinals $\alpha$, $\beta$, and $\gamma$. Use transfinite induction, and do not use the definition of "$\cdot$" (you won't need it.)
   2. Prove or disprove: $(\alpha + \beta) \cdot \gamma = \alpha \cdot \gamma + \beta \cdot \gamma$ for all ordinals $\alpha$, $\beta$, and $\gamma$.
3. Prove that the following statements are equivalent for every limit ordinal $\gamma$:
   1. $\alpha + \beta < \gamma$ for all $\alpha,\beta < \gamma$, and
   2. $\alpha + \gamma = \gamma$ for all $\alpha < \gamma$.
   Hint: use transfinite induction plus basic facts. Warning: many basic facts about natural numbers do not apply to all ordinals.
4. Prove that $\text{rank}(\bigcup x) \le \text{rank}(x)$ for every set $x$. Then give examples (with proof) where $\text{rank}(\bigcup x) < \text{rank}(x)$ and where $\text{rank}(\bigcup x) = \text{rank}(x)$.