Ordinal addition ($\alpha +\beta$)

Ordinal addition is ordered concatenation and is associative but generally not commutative.

For all ordinals $\alpha ,\beta ,\gamma$,This follows by transfinite induction on $\gamma$ from the recursive definition of ordinal addition.

Nonzero ordinals commute under addition precisely when they are positive finite right multiples of a common ordinal. This follows by comparing their Cantor normal forms at the first exponent where they differ.