Give the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent.

Which of the following are always true for ordinals $\alpha, \beta$ and $\gamma$ and which can be false? Give proofs or counterexamples as appropriate.

(i) $\alpha+\beta=\beta+\alpha$

(ii) $(\alpha+\beta) \gamma=\alpha \gamma+\beta \gamma$

(iii) $\alpha(\beta+\gamma)=\alpha \beta+\alpha \gamma$

(iv) If $\alpha \beta=\beta \alpha$ then $\alpha^{2} \beta^{2}=\beta^{2} \alpha^{2}$

(v) If $\alpha^{2} \beta^{2}=\beta^{2} \alpha^{2}$ then $\alpha \beta=\beta \alpha$

[In parts (iv) and (v) you may assume without proof that ordinal multiplication is associative.]