Ordinal addition is associative

For all ordinals $\alpha$, $\beta$, $\gamma$ (Ordinal (von Neumann)),

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

with $+$ as in Ordinal addition $\alpha +\beta$. Sums of ordinals may therefore be written without brackets, and this library does so from here on.

No choice principle is used. Associativity is not accompanied by commutativity, which is refuted among this page's false statements.

Why the order-type route rather than a recursion. Associativity can also be proved by transfinite induction on $\gamma$, and the limit case then needs the continuity 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 $1\cdot \beta =\beta$ together with the fact that $\{\beta +\xi :\xi \in \lambda \}$ is unbounded in $\beta +\lambda$. The order-type argument avoids the case analysis entirely: concatenation of well-orders is visibly associative, and $\alpha +\beta$ is the order type of $\alpha$ followed by $\beta$ transports that to the arithmetic.

Associativity does not rescue commutativity. The two are independent: the ordinals under $+$ form a semigroup with identity $0$ and nothing more. $1+\omega =\omega$ while $\omega +1={\omega }^{+}\mathrm{\ne }\omega$, which is computed in FALSE: ordinal addition is commutative; and left cancellation holds while right cancellation fails, since $0+\omega =1+\omega$ (FALSE: $\beta <\gamma$ implies $\beta +\alpha <\gamma +\alpha$).

Brackets are dropped from here on. Cantor normal forms such as ${\omega }^2+\omega \cdot 3+5$ (Cantor normal form: every nonzero ordinal is ${\omega }^{{\beta }_0}\cdot c_0+\cdots +{\omega }^{{\beta }_{k-1}}\cdot c_{k-1}$ with ${\beta }_0>\cdots >{\beta }_{k-1}$ and each $c_i$ a nonzero natural number, in exactly one way) are written unbracketed precisely because of this theorem.