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The inductive definition of ordinal addition isfor a limit ordinal $\lambda$.

The synthetic definition takes the order type of the disjoint union of a copy of $\alpha$ followed by a copy of $\beta$: every point of the first copy precedes every point of the second, and each copy retains its original total order.

Fix $\alpha$ and use transfinite induction on $\beta$. The synthetic sum with the empty order is $\alpha$. Appending a greatest element produces the successor ordinal rule. At a limit ordinal $\lambda$, the second copy is the union of its initial segments of types $\beta <\lambda$, so the whole order has type
${\sup }_{\beta <\lambda }(\alpha +\beta )$. Thus the synthetic operation satisfies the inductive recursion; uniqueness in transfinite recursion proves the definitions equivalent.

The set $\mathrm{WF}\subseteq {\omega }^{\omega }$ consists of the reals coding well-orders of $\omega$. For $x\in \mathrm{WF}$, the norm $\Vert x\Vert$ is the order type of the well-order coded by $x$.

A partial order in which the strict predecessors of each node form a well-order. The height of a node is the order type of its predecessors; nodes of a common height form a level. This order-theoretic meaning is distinct from a graph-theoretic tree.

For an infinite cardinal $\kappa$, the square principle supplies club sets $C_{\zeta }\subseteq \zeta$ at limit $\zeta <{\kappa }^{+}$, with order type at most $\kappa$, coherent at limit points: $C_{\gamma }=C_{\zeta }\cap \gamma$ whenever $\gamma$ is a limit point of $C_{\zeta }$. The bound precludes a single club set threading the entire sequence. For regular uncountable $\kappa$, coherence together with the weaker-looking size clause $\mathrm{cf}(\zeta )<\kappa \Rightarrow |C_{\zeta }|<\kappa$ implies the order-type bound. At $\kappa =\omega$, that clause alone is vacuous and cannot replace the bound.