ordinal arithmeticOrdinalArithmetic2003-02-23 18:52:042008-02-23 13:06:04Topic% this is the default PlanetMath preamble. as your knowledge
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%\PMlinkescapeword{theory}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:
\begin{itemize}
\item $x+0=x$
\item $x+Sy=S(x+y)$
\item $x+\alpha=\operatorname{sup}_{\gamma<\alpha} (x+\gamma)$ for limit ordinal $\alpha$
\end{itemize}
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:
\begin{itemize}
\item $x\cdot 0=0$
\item $x\cdot Sy=x\cdot y+x$
\item $x\cdot \alpha=\operatorname{sup}_{\gamma<\alpha} (x\cdot \gamma)$ for limit ordinal $\alpha$
\end{itemize}
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
\begin{itemize}
\item $\gamma+\alpha<\gamma+\beta$
\item $\gamma\cdot\alpha<\gamma\cdot\beta$
\item $\alpha+\gamma\leq\beta+\gamma$
\item $\alpha\cdot\gamma\leq\beta\cdot\gamma$
\end{itemize}