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 $
Title  ordinal arithmetic 

Canonical name  OrdinalArithmetic 
Date of creation  20130322 13:28:52 
Last modified on  20130322 13:28:52 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  7 
Author  Henry (455) 
Entry type  Topic 
Classification  msc 03E10 
Related topic  AdditivelyIndecomposable 
Related topic  CardinalArithmetic 