Ordinal addition

explain algebraic properties of ordinal addition

Nice — let’s unpack ordinal addition cleanly and with examples so the weird bits stick.

Definition (one convenient form)

Think of ordinals as order-types of well-ordered sets. The ordinal sum $\alpha+\beta$ is the order-type of the disjoint union of a copy of a well-order of type $\alpha$ placed before a disjoint copy of a well-order of type $\beta$. Equivalently, define by transfinite recursion on $\beta$:

• $\alpha+0=\alpha$.
• $\alpha+(\beta+1)=(\alpha+\beta)+1$ (i.e. the successor of $\alpha+\beta$).
• If $\lambda$ is a limit ordinal then $\displaystyle \alpha+\lambda=\sup_{\beta<\lambda}(\alpha+\beta)$.

This concatenation viewpoint explains most of the behavior below.

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Key algebraic properties

1. Identity

$0$ is a two-sided identity:

$$ 0+\alpha=\alpha,\qquad \alpha+0=\alpha. $$

2. Associativity

Ordinal addition is associative:

$$ (\alpha+\beta)+\gamma=\alpha+(\beta+\gamma) $$

(for all ordinals $\alpha,\beta,\gamma$). Intuitively concatenating three well-orders in either grouped order gives the same final ordering. (One proves it by transfinite induction on $\gamma$.)

3. Not commutative

In general $\alpha+\beta\neq\beta+\alpha$. Typical example:

$$ 1+\omega=\omega\quad\text{but}\quad \omega+1=\omega+1>\omega. $$

Putting a finite block before $\omega$ doesn’t change the order type, but putting it after does.

4. Monotonicity (how addition behaves in each argument)

• For fixed $\alpha$, the map $\beta\mapsto\alpha+\beta$ is strictly increasing: if $\beta<\gamma$ then $\alpha+\beta<\alpha+\gamma$. (So increasing on the right, and therefore right-cancellable — see below.)
• For fixed $\beta$, the map $\alpha\mapsto\alpha+\beta$ is nondecreasing but need not be strict: if $\alpha<\alpha'$ then $\alpha+\beta\le\alpha'+\beta$, but equality can occur. Example: $1<2$ yet $1+\omega=\omega=2+\omega$.

5. Cancellation

• Right cancellation holds: if $\alpha+\beta=\alpha+\gamma$ then $\beta=\gamma$. (This follows from strict increase in the right argument.)
• Left cancellation fails in general: $\beta+\alpha=\gamma+\alpha$ need not imply $\beta=\gamma$. Example: $1+\omega=\omega=0+\omega$ while $1\neq0$.

6. Continuity in the right argument (limits)

If $\lambda$ is a limit ordinal then

$$ \alpha+\lambda=\sup_{\beta<\lambda}(\alpha+\beta). $$

So addition is continuous from below in the right argument. It need not be continuous in the left argument. For instance $\sup_{n<\omega}(n+\omega)=\omega\cdot 2$, but $\left(\sup_{n<\omega} n\right)+\omega=\omega+\omega=\omega\cdot 2$ — okay that matches here — but there are other examples where limits on the left behave subtly; in short, continuity is the clean one on the right.

7. Successor behavior

For any $\alpha$, $\alpha+1$ is the successor of $\alpha$. In particular $\alpha+1$ is never a limit ordinal.

8. No general inverses / subtraction

There is no general subtraction: given $\gamma$ and $\alpha$ there need not exist $\beta$ with $\alpha+\beta=\gamma$. Even when such a $\beta$ exists it need not be unique unless you fix the left summand.

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Intuition & why the weirdness occurs

Because ordinal addition is concatenation of orders (put $\alpha$ before $\beta$), where position matters, you lose commutativity. The strictness in the right argument comes from the fact that inserting a strictly longer tail after the same prefix always yields a strictly larger ordering; but changing the prefix can be absorbed by long tails (finite prefixes get “swallowed” by $\omega$, etc.), so change on the left need not be visible.

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Quick summary (bullet)

• Definition: concatenation / transfinite recursion.
• Associative, identity $0$.
• Not commutative (e.g. $1+\omega\ne\omega+1$).
• Strictly increasing in the right argument → right-cancellative.
• Only nondecreasing in the left argument → left-cancellation may fail.
• Continuous in the right argument at limits.
• No general inverses (no group structure).

If you want, I can:

• show detailed proofs of associativity and right-monotonicity by transfinite induction,
• give a table of small examples (finite, $\omega$, $\omega+1,\omega\cdot 2,$ etc.), or
• contrast this with the natural (Hessenberg) sum which is commutative. Which would you like?