Ordinal addition $\alpha + \beta$

Let $\alpha$ and $\beta$ be ordinals (Ordinal (von Neumann)). The sum $\alpha + \beta$ is defined by recursion on $\beta$, in the three cases of Successor and limit ordinals:

$\alpha + 0 = \alpha, \qquad \alpha + \beta^{+} = (\alpha + \beta)^{+}, \qquad \alpha + \lambda = \bigcup\{\, \alpha + \beta : \beta \in \lambda \,\} \quad (\lambda \text{ a limit ordinal}).$

That exactly one operation satisfies these three clauses, and that all its values are ordinals, is Ordinal addition exists and is unique: the clauses at $0$, at a successor and at a limit determine one operation, and its values are ordinals, proved immediately above. The union in the limit clause is the least upper bound of the values already produced (claim (e) of Basic closure properties of ordinals), so it may be written $\sup$ and the clause read as "at a limit, take the supremum".

Notation. We write $1 = 0^{+}$, $2 = 1^{+}$, and so on for the finite ordinals, and $\sup$ for $\bigcup$ applied to a set of ordinals. The successor operation is now a special case of addition:

$\alpha + 1 = \alpha + 0^{+} = (\alpha + 0)^{+} = \alpha^{+},$

so from here on $\alpha + 1$ and $\alpha^{+}$ denote the same ordinal, and both notations are used, whichever reads better.

• The recursion is on the right argument only. The left argument $\alpha$ is a parameter, frozen before the recursion starts. This asymmetry is not an artefact of the presentation: ordinal addition is genuinely asymmetric, and the asymmetry is exactly what FALSE: ordinal addition is commutative and FALSE: $\beta < \gamma$ implies $\beta + \alpha < \gamma + \alpha$ exhibit later on this page.

• What the clauses say concretely. $\alpha + \beta$ is "$\alpha$, and then $\beta$ more steps". $\alpha + \beta$ is the order type of $\alpha$ followed by $\beta$ turns that picture into a theorem: $\alpha + \beta$ is the order type of a copy of $\alpha$ followed by a copy of $\beta$. The order-type description is usually the one to compute with; the recursion is the one that makes the definition legitimate.

• The first interesting value. $\omega$ is the least limit ordinal ($\omega$ is the least limit ordinal), and it is the least ordinal at which the third clause fires at all: below $\omega$ every ordinal is $0$ or a successor, so below $\omega$ this recursion is literally the Peano recursion for addition on $\mathbb{N}$. That the two agree there is On $\omega$ the ordinal $+$ and $\cdot$ are the Peano operations: $\omega$ is closed under ordinal $+$, $\cdot$ and exponentiation, and for naturals $m, n$ the ordinal $m + n$ and $m \cdot n$ are the natural-number sum and product, and it is a theorem, not a convention.

• Suprema need no completeness axiom. The limit clause takes the union of a set of ordinals, which is an ordinal and is their least upper bound, by claim (e) of Basic closure properties of ordinals and the remark following it. Nothing resembling the least upper bound property of $\mathbb{R}$ is assumed; it is a closure property of the ordinals themselves.

• Left addition of a limit is a limit. For a limit $\lambda$, the values $\alpha + \beta$ for $\beta \in \lambda$ are strictly increasing and their supremum is not attained, so $\alpha + \lambda$ is again a limit ordinal. This is recorded as a clause of Monotonicity of ordinal $+$ and $\cdot$: strictly increasing and continuous in the right argument, weakly increasing in the left, with left cancellation, and the identities $0 + \beta = \beta$ and $1 \cdot \beta = \beta$, where it is proved.