Re: Upstairs--Third Person narration

On Tue, 20 Mar 2007 15:50:44 -0400, Brian M. Scott wrote:

On Tue, 20 Mar 2007 16:38:45 +1000, "James A. Donald"
<jamesd@xxxxxxxxxxx> wrote in
<news:m1ruv2dksdjgpkb7lrqded69dlctqb155a@xxxxxxx> in
rec.arts.sf.composition:

James A. Donald:
But any statement made concerning [Conway's]
infinitesimals has a corresponding statement
concerning limits, which do not treat infinite
tasks as completed.

"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.

James A. Donald:
I do not think so: Give me an example.

"Brian M. Scott"
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.

The surreal number omega_1 is made from the infinite set
{1, 2, 3, 4, 5, ...| }.,

No: this is omega_0. It is born on day omega_0. The
ordinal number omega_1 is the first uncountable ordinal; it
is {{a : a is a countable ordinal} | }, and it is born on
day omega_1. No matter how it is represented as a left-set
/ right-set pair, the left set is necessarily uncountable.

[...]

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'.

An infinite strictly decreasing sequence of Conwey's
infinitesimals corresponds to the sequence of sequences
p[m]*2^(-n) where n and m independently approach
infinity, and p[m] is a strictly decreasing sequence of
computable numbers.

This is not a statement about limits of ordinary real
numbers, and I see no sense in which it corresponds 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:

The phrase "corresponds to" is not equivalent to the
phrase "is"

Irrelevant: you said that 'e is the number bounded below by
zero, and bounded above by the sequence 1, 1/2, 1/4, 1/8,
1/16 ......', which is false.

[...]

My conjecture, to which I have seen no counter examples,
is that any statement about infinite sets *corresponds*
to a statement about an infinite series of finite sets
that may grow without limit,

Your 'conjecture' is meaningless, since you have not defined
what you mean by 'corresponds to'.

[...]

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.

Reflect on what is meant by the word "corresponds"

You haven't given it any coherent meaning.

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.

Yes, but your are using limits, not infinitesimals.

Of course. You said that if we try to take that limit in
the surreal numbers, we'd be 'in for a world of hurt'; I
just showed that there was no problem at all.

What good are infinitesimals if we cannot do calculus
with them?

We can. I have.

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.

This does not in fact work,

Oh, it works, all right.

but a newsgroup is not the place to give the explanation,
particularly when we are having difficulty discussing
considerably simpler issues.

The problem is simply that you don't know very much about
the subject, and a good bit of what you think you know
appears to be wrong.

To be precise it works in a sense, you do
indeed have derivatives, but you then do not have
integrals, [...]

You're wrong. The same sort of thing with the standard part
map can be used to eliminate limits from basic integration
theory. There's even an undergraduate calculus text that
does exactly that (using hyperreals, not surreals).

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.

The epsilon delta definition tells us that if x is
within delta of the point, the function is within
epsilon of the limit. In order to actually do anything
useful with the epsilon delta definition you have to
apply it to a sequence of points that get ever closer to
the limit point,

*I* certainly don't. Bluntly, this is piffle.

so this statement, when actually used, is used about a
sequence.

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.

Or possibly just bull***.

You are using a different definition of order, one that
is not appropriate to discussing the differences between
operations on sets and operations on sequences.

Definitely bull***.

Brian

I love listening to mathematicians swearing at one another.

Regards,
Ric

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