(a) Give the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent. Give the inductive definitions of ordinal multiplication and ordinal exponentiation.

(b) Answer, with brief justification, the following:

(i) For ordinals $\alpha, \beta$ and $\gamma$ with $\alpha<\beta$, must we have $\alpha+\gamma<\beta+\gamma$ ? Must we have $\gamma+\alpha<\gamma+\beta$ ?

(ii) For ordinals $\alpha$ and $\beta$ with $\alpha<\beta$, must we have $\alpha^{\omega}<\beta^{\omega}$ ?

(iii) Is there an ordinal $\alpha>1$ such that $\alpha^{\omega}=\alpha$ ?

(iv) Show that $\omega^{\omega_{1}}=\omega_{1}$. Is $\omega_{1}$ the least ordinal $\alpha$ such that $\omega^{\alpha}=\alpha$ ?

[You may use standard facts about ordinal arithmetic.]