From: Colin Bartlett <colinb2r@...>
Date: 2010-12-29T04:53:53+09:00
Subject: Re: Numeric comparison with nil - Math masochists only!!

A pedantic and cautious (meaning I'm reasonably sure about what I
write below but I'm not 100% confident) person with a maths degree
from just over 30 years ago (so out of date, and I've forgotten a
lot!) writes:

On Tue, Dec 28, 2010 at 10:37 AM, Everett L Williams II
<rett@classicnet.net> wrote:
>>> On Sat, Dec 25, 2010 at 2:34 AM, serialhex<serialhex@gmail.com>  wrote:
>>>> ...
>>>> it returns -1 all the time so no matter what you compare it against it's
>>>> less than that (i mean, sereously, an empty set is WAAAAYYYYYYY less
>>>> than neg infinity, cause with neg infinity you still have SOMETHING right?)
>>>> ...
> I can't imagine where you got the idea that a nul is less than minus
> infinity. Mathematically, a null is basically undefined as in dividing by
> zero. I have no idea what your use of this is, but mathematically you are in
> the wrong pew.

1. In this case I think serialhex's "<=>" may not always be returning
the appropriate value for "comparisons" with nil (see below).
2. But for the general case, I remain to be convinced that a null is
*necessarily* undefined: depending on what one means by "null" in a
particular context, then it might make sense to compare something with
that. But if "a null is basically undefined" is restricting the use of
null to stuff that is undefined in a particular context, then
(cautiously) I agree!

3. The statement by serialhex that
  "an empty set is WAAAAYYYYYYY less than neg infinity"
did make me wonder if it was correct in the specific context of
Surreal numbers, and I have been looking at the articles. The basis
for comparisons is (in one formulation):
x <= y if and only if
(a) there are no values xLv in the Left set xL of x for which y <= xLv;
and
(b) there are no values yRv in the Right set yR of y for which yR <= x.

If the set xL is empty, then (a) is "vacuously" true; similarly, if
the set yR is empty, then (b) is "vacuously" true.
But the empty set itself is *not* (as I understand) it a Surreal
number, so strictly speaking I think (cautiously!) comparisons are not
with the empty set as such but with pairs of sets (L and R) of which
none or one or both of a pair may be the empty set.

In Tøndering's second chapter he does define comparisons between sets
of Surreal numbers, and then states:
  "It follows from the definition above that
      EmptySet <= b
   is always true, regardless of the value of b; and so is
      EmptySet > b, etc"

One needs to be very careful: I think (cautiously) that the definition
of Order in the Wikipedia article is correct, but that the "friendly"
explanations in brackets after the proper parts of the definition are
(probably/possibly!) either misleading or wrong.
http://en.wikipedia.org/wiki/Surreal_number