Bill Page wrote:
>
> Thanks Waldek. I don't really have a problem with providing a value
> for x/0. This is not so different than considering the CardinalNumber
> domain as an extension of NonNegativeInteger. CardinalNumber is not
> used much in FriCAS except as the domain of the operation 'dimension'
> (of VectorSpace, etc.) however it includes a value of Aleph(0) for
> infinite cardinality. I suppose that it would be possible to define
> something like this for Float, e.g.
>
> x/0 = Aleph(1)
>
> and we have
>
> x ~= Aleph(1)
>
> for all values in Float except Aleph(1).
>
Bill, I am afraid you did not notice the main point: in classical
logic we want total functions. If you add sometning like Aleph(1)
you need to define all field operations on Aleph(1). If you add
a single element then there is _no way_ to extend operations
so that field axioms are satified. If you add more elements
than what you are doing is effectively replacing your original
field by a bigger one and than choosing value for '1/0' _in this
bigger field_. In other words, to make '/' into a total function
you have to choose value for '1/0' _inside_ the field.
> But I worry if the proposal really is that
>
> x/0 = 0
>
> since that could do a lot of damage to many other desirable properties that
>
> Float has Field
>
> should imply, e.g. as Bertfried pointed out concerning it's
> topological properties.
Float is bad example because strictly speaking it is not a field
(operations are nonassociative). However defining 'x/0' does not
really change properties of field: normal field axioms only say
what happens when you divide by nonzero element.
--
Waldek Hebisch
[email protected]
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