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

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

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

(ii) If $\alpha$ and $\beta$ are uncountable then $\alpha+\beta=\beta+\alpha$.

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

(iv) If $\alpha$ and $\beta$ are infinite and $\alpha+\beta=\beta+\alpha$ then $\alpha \beta=\beta \alpha$.