Bill Page wrote:
>
> Waldek,
>
> Thanks for your patient explanation. So we define
>
> 1/0 = 0
>
> but we do not call this a multiplicative inverse. Should we say then that
>
> 0^(-1)
>
> is still undefined?
>
What is defined depends on exact definitions in use. If field
axioms use inverse denoted by x^(-1), then of course 0^(-1)
is defined. If field axioms use '/' then 'x^(-1)' is probably
undefined and then also '0^(-1)' is undefined. In the constructor
I posted only '1/0' is defined, for '0^(-1)' one would have to add
extra code.
One more remark: we are talking about "formal systems" here.
There are no deep meaning here, like in case of empty set formal
definitions may be sligthly different than intuition, but are
made so that formal systems works well for specific purpose
(in our case for "first order" theory of fields). If you
look for meaning and want to allow '1/0' you will probably use
different mathematical structure (like projective plane), which
however is no longer a field.
--
Waldek Hebisch
[email protected]
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