Give the inductive and the synthetic definitions of ordinal addition, and prove that they are equivalent. Give an example to show that ordinal addition is not commutative.

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 limit ordinals then $\alpha+\beta=\beta+\alpha$.

(iii) If $\alpha+\beta=\omega_{1}$ then $\alpha=0$ or $\alpha=\omega_{1}$.

(iv) If $\alpha+\beta=\omega_{1}$ then $\beta=0$ or $\beta=\omega_{1}$.