Re: Motion 0035 -- New levels 1 and 1a
Nate Hayes wrote:
> If f(X) is the natural interval extension of a real function f on X,
> then for any interval extension F(X), we therefore have for FTIA:
> f(X) \subseteq F(Upsilon(Z)).
> And likewise for the range of F(X) we have:
> f(X) \subseteq Upsilon(Omega(F(X)))
> (c. f. Proposition 1) which makes it safe to interpret any interval
> extension of F, generally, as
> Omega(F(Upsilon(Z))). (*)
Ok: we really are dealing with the current Level 1 (NOT the new one
that only supports nonempty bounded intervals). Level 1a objects
are converted to IRbar objects by Upsilon, operated on the way we
have had it for the past few years, then converted back to Level 1a
by means of Omega.
Alexandre Goldsztejn raised the same issue I did on the two-sorted
domain of Omega, and Nate agreed that Omega([a,b]) should be the
singleton { [a,b] }, so that Level 1a objects are always sets of
bounded intervals (or, if Omega(Empty) is the singleton { Empty },
non-empty sets of empty or bounded intervals).
So we have a new Level 1 that is never used, and a Level 1a whose
objects are never used directly. The objects that are really being
used are always Upsilon(Omega(object that is actually used)) --
namely the members of IRbar as we have been using them all along.
(This of course solves my quandary as to how Union and Intersection
would be defined: NOT on the Level 1a objects themselves, but on
their Upsilon images.)
This is all conceptual of course. The real work happens at Level 2,
which is presumably the real and only purpose of Level 1a objects,
by defining how these are mapped to elements of a finite subset of
the Level 1a domain -- which are in one-to-one correspondence with
the elements of IRbar that constitute the domain of the old Level 1.
As Alexandre said: "So what is the point?"
Michel.
P.S. I wonder how industrial-minded people would react to this
mathematical banter? I would expect them to prefer a
presentation that keeps the mathematical foundations mostly
out of sight. It took several generations of mathematicians
to become comfortable with the element/subset distinction...
and here it becomes front and center!
---Sent: 2012-05-09 22:49:51 UTC