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LSE

Nate Hayes a écrit :

John Pryce wrote:

First let me say I agree with criticisms I've seen of the Motion 13
Comparisons text in §5.6.9. They *need* definitions from first principles 
in terms of sets, rather than endpoints. If we don't have that then how
they act on empty and unbounded inputs is likely to be ad-hoc and possibly 
contradictory.


I remember the discussions surrounding this issue during the Motion 13
discussion period. Before too much time is invested rehashing the same
topic, I would recommend re-reading some of those threads.

In a nutshell, the issue as I recall boiled down to "unbounded" vs.
"overflown" intervals. To paraphrase Arnold Neumaier's distinction between 
these two concepts:

   -- The "unbounded" interval [7,+Inf] is a single interval (set of real
numbers) with an upper endpoint that is not bounded.

   -- The "overflown" interval [7,+OVR] on the other hand is a family of
intervals. There are an infinite number of intervals in the family, but each 
element of the family is closed and bounded.
I disagree. An interval such as [7,+OVR] refer to A SINGLE UNKNOWN interval in he family.
The correct definitions and/or interpretations of the comparison operations 
depend on these distinctions. For example, from a purely mathematical
perspective the comparison of unbounded intervals
   [7,+Inf] \interior [1,+Inf]

Thus [7,+OVR] \interior [1,+OVR ] is UNKNOWN not UNORDERED


While   [1,+OVR ]  \interior  [7,+OVR]   is always FALSE
since no interval of the [1,+OVR] family is interior of of any interval of the [7,+OVR] 

Dominique







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