Re: What does "infinty as number" mean?
Nate et al,
My understanding it that we are working with R* when
"infinity is a number," and we are working with R when
"infinity is not a number." The practical consequence,
if Cset theory is used, is that
0* Inf = [-Inf,Inf]
if infinity is a number
and
0 * Inf = 0
(in, say multiplication of intervals involving
the end points) if infinity is not a number.
We may argue about this even if Cset theory is not used.
Furthermore, there are controversial consequences of using
Cset theory that are present in both cases. For instance,
in the Vienna proposal, one sees the following
example:
[0,1] / [0,1] = [-Inf,Inf]
if infinity is a number, while
[0,1] / [0,1] = [-Inf,Inf]
if infinity is not a number. It is argued in the Vienna
proposal that most, if not all, rigorous computations
can use [0,1] / [0,1] = [0,Inf) or [0,Inf}, respectively,
since we want to describe the set of all values with
the values taken only from the intervals in question.
The problem is that, with
Cset theory, the interval value is determined to be the
union of the point values, and the point value 0/0, as
the set of all limits x/y as x and y go to zero independently,
is [-Inf,Inf] or (-Inf,Inf), respectively. John Pryce has
proposed modifications to Cset theory to allow one-sided
limits, or limits only with values from the intervals only.
I hope this clarifies things, and helps us to decide these
issues. It seems we want to define an interval operation
to be the set of all results with values taken from the intervals.
The problem is when the operation is not defined for particular
values, such as 1/0. Cset theory is consistent in the sense
that [1,1] /[0,0] is defined and fits naturally into the
scheme of things, regardless of whether we consider it to
be [-Inf,-Inf] U [Inf,Inf] (if infinity is a number), or
\emptyset (if infinity is not a number). However, as is
pointed out in the Vienna proposal, this gives overestimations
in many computations. So, do we want a clear specification
in terms of the mathematics of limits, or do we want sharper computations,
but maintaining rigor if we know what we are doing?
Best regards,
Baker
On 3/1/2009 8:25 AM, Nate Hayes wrote:
Dear Members,
What does the expression "infinity as number" mean to this 1788 group?
For better or worse, I had been under the impression it means the
following three things:
1. Infinities are not members of the interval, although they may appear
as endpoints of an interval to indicate an open (unbounded) endpoint.
2. To evaluate the difference (Inf-Inf) or ratio (Inf/Inf) of two
infinities, the infinities are replaced with a real number x and then
the arithmetic operation is considered in the limit as the magnitude x
tends towards infinity. The same is true for 0*Inf, which leads to 0*Inf=0.
3. Because of 2), Inf-Inf=0, Inf/Inf=1, and 0*Inf=0, but other
arithmetic operations involving infinities are as usual, such as
Inf+a=Inf for any real number a.
Did I abuse the terminology too much?
Sincerely,
Nate Hayes
Sunfish Studio, LLC
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R. Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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