Bill Page wrote:
>
> Thanks Ralf, that makes a bit more sense. What exactly (formally) does
> one get if one requires x/0 = 0? It is not a field or even a division
> algebra, right?
>
Well, x/0 = 0 is pretty standard in logic: in logic one wants
total functions so _some_ definition of x/0 is required. Then
using 0 as a value is most natural. If you want to stay in
the "most classical logic", that is first order logic with
total functions, then FriCAS field is not a legitimate mathematical
structure, becase in FriCAS division is a partial function
(so for FriCAS you need "stronger" logic).
The intent of this definition is that you really do not want to
compute things like 1/0, but for formal reasons you can not avoid
them. Consider:
(x ~= 0) and (z = 1/x)
This is a fine, sensible logical formula: when 'x' is 0 then the
formula is false due to first term and if 'x' is different from 0,
then division is well-defined. But if 'and' is a function and
we evaluate arguments first we may be forced to compute '1/0',
use it in comparison and then effectively throw out the result
of the comparison.
Of course, if one really makes use of exact value of x/0 then
it is confusing...
--
Waldek Hebisch
[email protected]
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