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) $\alpha \beta=\beta \alpha$.

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

(iii) If $\alpha \geqslant \omega^{2}$ then $\alpha+\omega^{2}=\omega^{2}+\alpha$.

(iv) If $\alpha \geqslant \omega_{1}$ then $\alpha \omega_{1}=\omega_{1} \alpha$.