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)
-
•
x+α=supγ<α(x+γ) for limit ordinal
α
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,
ω+1=ω+S0=S(ω+0)=Sω |
but
1+ω=supn<ω1+n=ω |
Multiplication in turn is defined by iterated addition:
-
•
x⋅0=0
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•
x⋅Sy=x⋅y+x
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•
x⋅α=supγ<α(x⋅γ) for limit ordinal α
Once again this definition is equivalent to normal multiplication when x and y are finite, but is not commutative:
ω⋅2=ω⋅1+ω=ω+ω |
but
2⋅ω=supn<ω2⋅n=ω |
Both these functions are strongly increasing in the second argument and weakly increasing in the first argument. That is, if α<β then
-
•
γ+α<γ+β
-
•
γ⋅α<γ⋅β
-
•
α+γ≤β+γ
-
•
α⋅γ≤β⋅γ
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 |