Incorrect. This series converges only when i=0 and diverges for all
other i.
Not incorrect. Ross is saying the same thing you are but more
specifically. Standard mathematics considers a divergent series simply
that. Ross is comparing this sum to a standard infinite sum, a standard
infinite unit.
Even when you take infinite/infinitesimals into account, you can still
have a divergent series. Essentially, it means that the series grows
bigger than even the unlimited (infinite) hyper-reals.
As for the concept of "a standard infinite unit", you run into a
problem since for any infinite hyper-natural n, n-1 is also an
infinite hyper-natural. Just as there is no largest finite number N
that you can make infinite by adding 1, there is no smallest infinite
hyper-natural M that you can make finite by subtracting 1. I suppose
you could choose some hyper-natural N* and call that your "standard
infinite unit", but the selection is arbitrary, and there will always
be an infinite number of hyper-naturals which are smaller (and larger)
than this "standard", irrespective of choice.
Where you have a standard infinite unit I, such as the number of
reals in the unit interval, then you also have a standard infinitesimal
unit i, the space that each of those reals occupies in that interval,
the multiplicative inverse of I. I*i=1.
The reverse case will exist with this "standard infinitesimal unit i":
there will always be a smaller (and larger) infinitesimal. If i is
any positive infinitesimal, then i/2 is a smaller one, whilst 2i is a
larger one. Designating one as a "standard" is arbitrary, and I am
unsure as to what that achieves.
As for designating "the number of reals in the unit interval" as your
"standard infinite", this number is larger than all the hyper-reals,
and thus is larger than the inverse of any infinitesimal, no matter
how small an infinitesimal we choose to invert. This infinite size is
so large, it is in another class of numbers completely: the cardinal
numbers.
Hi Jonathan -
Do you think it unreasonable to consider an uncountable unit I as the
number of reals in (0,1], and an infinitesimal unit i as its reciprocal?
Doesn't this marry measure and count, the infinite with the finite, and
infinitesimal, in a relatively consistent way?
Tony Orlow
Hi Tony,
Although there is a great deal of appeal attempting to marry these two
concepts (cardinality and measurability), the operations acting upon
them are very different.
For example, consider the "+" symbol, which we use for addition. In
Measure Theory, + is used for the summing of lengths of the union of
sets, is increasing (that is x+y>x,y), and is limited to countable
additivity. In Cardinality, + is used for 1-1 correspondence of the
union of sets, is "flat" (that is, x+y=max(x,y) for infinite x,y) and
allows for uncountable additivity. We use the + symbol in both
instances, but they are referring to entirely different things.
To see this, let I0 = [0,1) and I1= [1, 2). The measures of I0 and I1
are 1 each, and so m(I0 U I1) = m(I0) + m(I1) = 1 + 1 = 2. However,
the cardinalities are each c, and we have |I0 U I1| = |I0| + |I1| = c
+ c = c.
The problem becomes compounded with multiplication. You want to have
some infinitesimal i that is the multiplicative inverse of the
cardinal number c. But there is no multiplicative inverse on a set
whose multiplication is "flat". Any such value i will quickly
generate a contradiction:
1 = c*i = (c+c)*i = (c*i + c*i) = (1 + 1) = 2
Hyper-reals are essentially the largest numbers we have which behave
in a multiplicatively invertible way.
Regards,
Jonathan Hoyle
Eastman Kodak
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