 Solution

The inductive definition of ordinal addition isfor nonzero limit ordinals $\lambda$. The synthetic definition is the order type of a disjoint copy of $\alpha$ followed by a disjoint copy of $\beta$, retaining the order within each block and declaring every element of the first block smaller than every element of the second.

These are equivalent by transfinite induction in the second argument. An empty second block leaves $\alpha$; a successor block appends one final point; and a limit block is the increasing union of its proper initial blocks, whose order types have the indicated supremum. Thus the synthetic construction satisfies exactly the inductive recursion, which determines the operation uniquely. It is not commutative: $1+\omega =\omega$, whereas $\omega +1>\omega$.