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Re: Cancellation and Level-1 overflowed intervals

On 04/26/2012 05:47 PM, Michel Hack wrote:

Nate's main argument is, if I interpret correctly, the following:

Date: Wed, 11 Apr 2012 10:03:43 -0500
...
For me, it is more important that the Level 1 arithmetic is cancellative,
and the current P1788 model with unbounded intervals does not have this
property.


Yet I fail to see how      A + X = B + X
can imply                  A = B
when                       X = [x, OVR]
even when we interpret the latter to denote a family of intervals.
requiring cancellation for interval arithmetic with unbounded intervals is like requiring cancellation for real addition on the completed reals. It it simply impossible in both cases to extend the cancellation law to the more comprehensive situation. But nobody thinks the arithemtic on the real completion defective because of that.
In the same vain, that cancellation fails on the completed intervals is immaterial. No algorithm I know of depends on this property. 


Arnold Neumaier