 Solution

The inductive definition of ordinal addition is the transfinite recursionfor every nonzero limit ordinal $\lambda$. The synthetic definition takes disjoint well-orders $A$ and $B$ of order types $\alpha$ and $\beta$, puts every member of $A$ before every member of $B$, and defines $\alpha +\beta$ to be the order type of this ordered sum.

These definitions agree by transfinite induction on $\beta$. The empty second order contributes nothing; adjoining a greatest element gives the successor clause; and a limit well-order is the union of all of its proper initial segments, so its ordered sum with $A$ has order type ${\sup }_{\gamma <\lambda }(\alpha +\gamma )$.