Waldek, On Fri, Mar 25, 2011 at 1:01 PM, you wrote: > 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. >
I think I do understand the main point. I do not think it is possible to do this _inside_ the field of real numbers. Any way you do this the result cannot be a Field (or even a Ring), however it might still be useful http://en.wikipedia.org/wiki/Real_projective_line where Aleph(1) = - Aleph(1) Or http://en.wikipedia.org/wiki/Extended_real_number_line where there are both + and - infinities. >> 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). True. I assumed that we were already glossing over that issue. > However defining 'x/0' does not > really change properties of field: normal field axioms only > say what happens when you divide by nonzero element. If the inverse of 0 is 0 then the domain should not even be a Group let alone a Field. But I think your example of how to define such a domain in FriCAS is still interesting. I am curious why it seemed to be more difficult to do this than one might have expected. Are there some hidden assumptions somewhere in FriCAS about Float? Regards, Bill Page. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/fricas-devel?hl=en.
