ordinal arithmetic

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:

• $x+0=x$

• $x+Sy=S(x+y)$

• 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

Multiplication in turn is defined by iterated addition:

• $x\cdot 0=0$

• $x\cdot Sy=x\cdot y+x$

• 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

Both these functions are strongly increasing in the second argument and weakly increasing in the first argument. That is, if then

• $\alpha +\gamma \le \beta +\gamma$

• $\alpha \cdot \gamma \le \beta \cdot \gamma$