Waldek,
On Mon, Mar 28, 2011 at 12:54 PM, you wrote:
> Bill Page wrote:
>> 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.
But presumably by definition x^(-1) denotes the multiplicative inverse
x * x^(-1) = 1
and 0^(-1) is not such an inverse. It is interesting, maybe that
FriCAS gives different error messages for these two cases:
1.0/0.0
and
(0.0)^(-1)
> 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.
>
What value would you give it?
> 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.
>
Since FriCAS is a computer *algebra* system and not a automated proof
system it is not clear to me in what sense FricCAS currently supports
a first order theory of fields but of course one can program FriCAS in
a general manner for many different purposes.
As I suggested earlier my preference would definitely be to consider a
domain representing the projective plane (unsigned infinity) or maybe
a two point compactification (signed infinity) rather than one in
which we have a/0=0.
Regards,
Bill Page.
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