Re: Upstairs--Third Person narration

On Mon, 19 Mar 2007 14:47:28 +1000, "James A. Donald"
<jamesd@xxxxxxxxxxx> wrote in
<news:uq1sv25oe3on7au66uasba6v10dct5epke@xxxxxxx> in
rec.arts.sf.composition:

On Sun, 18 Mar 2007 21:59:22 -0400, "Brian M. Scott"
But any statement made concerning infinitesimals has
a corresponding statement concerning limits, which
do not treat infinite tasks as completed.

On Sun, 18 Mar 2007 21:59:22 -0400, "Brian M. Scott"
This isn't actually true: there are statements about
infinitesimals -- in any of the systems of making them
rigorous that I've encountered -- that do not
correspond to statements about limits of ordinary real
numbers.

I do not think so: Give me an example.

The surreal numbers include all of the ordinals; in
particular, omega_1, the first uncountable ordinal, is a
surreal number. Let e_1 be its reciprocal. Of course e_1
is infinitesimal. But no countable set of real numbers
suffices to construct e_1.

I doubt that with any meaningful notion of 'corresponds to'
you can come up with a statement about limits of ordinary
real numbers corresponding to the statement 'there is an
infinite, strictly decreasing sequence of positive
infinitesimals'.

Conway's definition of an infinitesimal corresponds
directly to the equivalent statement about limits: For
Conway, the simplest infinitesimal is e, which is equal
to 1/w, e is the number bounded below by zero, and
bounded above by the sequence 1, 1/2, 1/4, 1/8, 1/16
......

This is simply wrong. First, the left and right sets of e
are *sets*, not sequences: e_L = {0}, and e_R = {1/2^n : n
in N}. Secondly, e is not *the* surreal number
satisfying 0 < e < 1/2^n for each n in N, it's merely the
first one constructed (in Conway's original version, which
is not the only way to get the surreal numbers).

Which sounds very much like what people do when they
construct calculus by taking limits.

Not if one understands both ordinary limits and Conway's
construction; they have nothing to do with each other. In
fact surreal and hyperreal numbers allow one to do calculus
*without* taking limits.

So any statement involving e maps to a statement about
sequences.

I'm afraid not.

Further, when one takes sequences, one can easily show
that the limit of ((x+e)^2 - x^2)/e approaches 2x as e
approaches zero.

I trust that e here is not the infinitesimal above: that's a
fixed (surreal) number, so it can no more approach 0 than
1/4 can approach 0.

If, however, one tries to do the equivalent with surreal
numbers, one is in for a world of hurt.

Not at all. Let f(x) = x^2, and let u be any positive
surreal number. Then for any surreal v such that 0 < |v| <
u we have |f(x+v) - f(x)|/v - 2x| = |v| < u, so if we
simply extend the usual epsilon-delta definition of limit to
the surreals, we get that the limit of (f(x+v) - f(x))/v as
v approaches 0 is 2x.

But there's actually no need for limits. Given a real
function f and its surreal extension F, if there is a real
function g such that

(F(x+u) - F(x))/u - g(x)

is infinitesimal for each non-zero infinitesimal u, then g
is unique, and we *define* f' to be g. It's then possible
to show that this is equivalent to the usual
definition, but there are no limits at all in this version.

By treating infinity as actually completed, one has lost
the information revealed by considering the process of
travelling towards infinity.

There is no 'travelling' in the usual epsilon-delta
definition of limit. And in the usual definition of the
limit of a function at a point there are no sequences,
either.

In order to actually do calculus using something like the
surreal numbers, one has to reintroduce the order
relationship.

One can't do ordinary calculus on the reals without dealing
with the usual ordering of the reals, and in any case the
ordering of the surreals is inherent in their
construction. (In fact, it's a primitive in the Alling
axiomatization.)

Since a set has no order, an infinite set requires us to
treat infinite tasks as already completed, but as soon as
one constructs an order relationship, one has ceased to
treat infinite tasks as completed

This is simply false.

The fact that we are treating infinite tasks as completed
is actively stopping us from doing calculus with Conway's
surreals. Nothing has been gained, and something has
been lost.

The notions of differentiation and derived function
generalize to the surreal numbers, though I believe that
there is as yet no satisfactory generalization of
integration. (Note that doing calculus on the surreals is a
completely different matter from using the surreals to do
calculus on the reals.)

[...]

It isn't, however: any recursive construction of
length greater than omega requires treating an
infinite task as completed.

Since the continuum hypothesis can be true or false,
according to preference, one cannot actually discover
anything interesting or useful from the the kind of
facts that presuppose that the continuum hypothesis is a
concrete question with a meaningful answer.

Recursive constructions of length greater than omega
frequently have nothing to do with CH. And whether
something is interesting or useful is a matter of opinion,
and neither 'interesting' nor 'useful' is a one-place
predicate. 'Interesting' is at least a three-place
predicate: x is interesting to y in context z. 'Useful' is
at least a two-place predicate: x is useful in doing y.

Treating infinity as uncompleted excludes a great deal
from discussion, but it is unclear that what it excludes
has any meaning.

Unclear to you, perhaps; it's not at all unclear to me.

[...]

Brian
.

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