I wrote:
>
> Ralf Hemmecke wrote:
...
> > usually would like to write + even when dealing with ordinals. What
> > exactly is the semantics of your +? It's a commutative one, which
> > doesn't exist in the ordinals.
>
> See books on set theory. Unfortunately, top Google hits are to
> books and I saw no short online source.
>
Ok, a little crash course on ordinal arithmetic. Every ordinal
alpha has unique finite expansion in base omega:
alpha = \sum c_i omega^{a_i}
where c_i are positive integers and a_i is decreasing sequece of
ordinals. Here power is defined inductively using relation
omega^{a+1} = omega^a * omega
where * is ordered multiplication. The sum above is with respect
to ordered addition.
We define natural addition by summing expansions above like
polynomials. We define natural multiplication by multiplying
expansion above like polynomials and using natural addition
for exponents.
One can check that in the expansion above one can replace
ordered addition by natural addition (performing operations
in specified order we get the same results). Also, in
definition of omega^a one can replace ordinal multiplication
by natural multiplication (one can define power with different
base and then they differ, but base omega is special).
As you see natural operations on ordinals lead directly to
given implementation.
BTW: The adjective "natural" is not my invention, it is
established terminology. I concoted the term
"ordered addition" and "ordered multiplication" to make
explicit distinction between natural operations and
operations defined using theory of orders.
--
Waldek Hebisch
[email protected]
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