ordinal arithmetic
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:
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for limit ordinal
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,
but
Multiplication in turn is defined by iterated addition:
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for limit ordinal
Once again this definition is equivalent to normal multiplication when and are finite, but is not commutative:
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Both these functions are strongly increasing in the second argument and weakly increasing in the first argument. That is, if then
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Title | ordinal arithmetic |
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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 |