= Solution
The inductive definition of <ordinal addition> is
$$
\alpha+0=\alpha,\qquad\alpha+(\beta+1)=(\alpha+\beta)+1,
\qquad\alpha+\lambda=\sup_{\beta<\lambda}(\alpha+\beta)
$$
for 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: \b[$1+\omega=\omega$, whereas $\omega+1>\omega$].
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