Ordinal arithmetic is the extension of normal arithmetic to the transfinite ordinal numbers. The successor operation $Sx$ (sometimes written $x+1$ although this notation risks confusion with the general definition of addition) is part of the definition of the ordinals, and addition is naturally defined by recursion over this:

• $x+0=x$
• $x+Sy=S(x+y)$
• $x+\alpha=\operatorname{sup}_{\gamma<\alpha} (x+\gamma)$ for limit ordinal $\alpha$

If $x$ and $y$ are finite then $x+y$ under this definition is just the usual sum, however when $x$ and $y$ become infinite, there are differences. In particular, ordinal addition is not commutative. For example, $$\omega+1=\omega+S0=S(\omega+0)=S\omega$$ but $$1+\omega=\operatorname{sup}_{n<\omega} 1+n=\omega$$

Multiplication in turn is defined by iterated addition:

• $x\cdot 0=0$
• $x\cdot Sy=x\cdot y+x$
• $x\cdot \alpha=\operatorname{sup}_{\gamma<\alpha} (x\cdot \gamma)$ for limit ordinal $\alpha$

Once again this definition is equivalent to normal multiplication when $x$ and $y$ are finite, but is not commutative: $$\omega\cdot 2=\omega\cdot 1+\omega=\omega+\omega$$ but $$2\cdot\omega=\operatorname{sup}_{n<\omega} 2\cdot n=\omega$$

Both these functions are strongly increasing in the second argument and weakly increasing in the first argument. That is, if $\alpha<\beta$ then

• $\gamma+\alpha<\gamma+\beta$
• $\gamma\cdot\alpha<\gamma\cdot\beta$
• $\alpha+\gamma\leq\beta+\gamma$
• $\alpha\cdot\gamma\leq\beta\cdot\gamma$