PlanetMath: ordinal arithmetic

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ordinal arithmetic

(Topic)

Ordinal arithmetic is the extension of normal arithmetic to the transfinite ordinal numbers. The successor operation (sometimes written , 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:

$x+\alpha=\operatorname{sup}_{\gamma<\alpha} (x+\gamma)$ for limit ordinal $\alpha$

If and are finite then under this definition is just the usual sum, however when and become infinite, there are differences. In particular, ordinal addition is not commutative. For example,

$\displaystyle \omega+1=\omega+S0=S(\omega+0)=S\omega$

but

$\displaystyle 1+\omega=\operatorname{sup}_{n<\omega} 1+n=\omega$

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 and are finite, but is not commutative:

$\displaystyle \omega\cdot 2=\omega\cdot 1+\omega=\omega+\omega$

but

$\displaystyle 2\cdot\omega=\operatorname{sup}_{n<\omega} 2\cdot n=\omega$

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

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

"ordinal arithmetic" is owned by Henry.

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See Also: additively indecomposable, cardinal arithmetic

Attachments:: properties of ordinal arithmetic (Result) by CWoo
ordinal exponentiation (Definition) by CWoo

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Cross-references: weakly increasing, argument, strongly increasing, functions, equivalent, multiplication, commutative, differences, infinite, sum, finite, limit ordinal, addition, operation, successor, ordinal numbers, arithmetic, normal, extension
There are 3 references to this entry.

This is version 4 of ordinal arithmetic, born on 2003-02-23, modified 2008-02-23.
Object id is 4052, canonical name is OrdinalArithmetic.
Accessed 4504 times total.

Classification:

03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

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