ordinal arithmetic

Ordinal arithmetic is the extension of normal arithmetic to the transfinite ordinal numbers. The successor operation $S x$ (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:

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$x+0=x$
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$x+Sy=S(x+y)$
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$x+\alpha=\operatorname{sup}_{\gamma<\alpha}(x+\gamma)$ 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:

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

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

$\alpha\cdot\gamma\leq\beta\cdot\gamma$

Title	ordinal arithmetic
Canonical name	OrdinalArithmetic
Date of creation	2013-03-22 13:28:52
Last modified on	2013-03-22 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