Re: Calculus XOR Probability

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:

<snip>

When you say there's a "problem" with my limit, what do you mean?

I mean that it doesn't take into account anything but location, and you're
using it to measure distance.


Distance and length are real numbers.

Is lim n->oo {C_n} a real number, or is it a set of points?

It's a sequential set of points, that is a line of some sort, with a real
measure called length.

No, it's not.

{C_n} is a /sequence/ of /sets of points/.

C_n for n in N is a sequence of sequential sets of elements, which can either
be the corners of the steps, or the line segments which connect them.


A sequence of elements from some set X is essentially a function f : N
-> X. We write "f_n" instead of f(n), so we don't get confused and
think that f(3/2) might have some meaning. We then write {f_n} to
indicate the whole function, rather than its value at some particular
n.

(lim n->oo {C_n}) is defined to be a particular /set of points/. There
is no requirement that this limit be "a line of some sort, with a real
measure called length".

If that is the type of limit you're using, you have no business claiming it
gives a valid measure, then blaming the fact that it doesn't on infinite
induction.

First: Tony, could you do us a favour? Quit the "your system" stuff? I
think it's fair to say that Chas has not invented _anything_ above (or
below); he's simply trotting out standard definitions that can be found
in any elementary text on analysis. For example, in my ancient copy of
Apostol, Chapter 4 starts on page 61 with a generalised definition of
limit. Since Chas says he gave you a definition of limit before but I
can't see in this post, I'll add Apostol's here. I warrant that there
will be no problem, because actually all mathematicians use the same
definition:

[Forgive asciification problems: e for epsilon, d for delta]
------------------- QUOTE Tom Apostol ---------------------
4-1 Definition. The symbolism

lim[x->x_0] f(x) = A

means that for every number e>0 there is another number d>0 such that
whenever
0 < |x-x_0| < d, then |f(x)-A| < e.

In the case of sequences, lim[n->oo]x_n = A means, intuitively, that
the terms x_n of our sequence will be as near to A as desired, provided
only that n is sufficiently large. A more rigorous definition takes the
form:

4-2 Definition. We write

lim[n->oo] x_n = A

to mean that for every number e>0 there is an integer N such that
whenever n > N, then |x_n-A| < e.
----------------- END QUOTE Tom Apostol ---------------------

Now the point is that this definition can be applied also to the limit
of a sequence of sets of points in R2, which is what Chas has done (it
means you need a metric on the sets of points, as Chas also explained,
though I think it went past you). This is what all mathematicians mean
by "limit". If you honestly aspire ever to convince any mathematician
of anything at all, you will not use the term "limit" to mean something
completely different that you think you have invented.

Can you provide a definition of what it means for a set C to be "a
line of some sort, with a real measure of length"?

E A (endpoints)
E B
E x<>A -> E y: x=successor(y)
E x<>B -> E y: y=successor(x)
x=successor(y) <-> y=predecessor(x)
Length=Sum(x=successor(A)->B: x-predecessor(x)) where subtraction of points
indicates distance, which depends on the dimension of space.

Can you /then/ show how /my definition/ of lim n->oo {C_n} satisifies
/your definition/ of "a line of some sort, with a real measure of
length"?

Did you read Chas's question carefully? Can you show how /my
definition/ of the limit satifies /your definition/ of this line
thingie? (Here "/my definition/" of course is the same as Chas's, since
it the mathematical one.)

Takin the corners of the stairs as your sequence of points, with n stairs, each
is 1/n in height and width, and when you sum all those up, you get 2.

I didn't actually see the definition of C_n, but for any (pofnat) n,
C_n is a "staircase curve", a set of points in R2, complying with the
definition (Chas gave) for a 'curve' (viz. a mapping from R to R^2). So
no, you have not shown the required satisfaction-of-definition. Try
again.

<snip>

Okay, but what happens in the limit, in your estimation? When the distances
between successive points become less than any real number, you seem to think
that the distance is zero or something.

There's no need to ask for Chas's personal "estimation". Chas can read
definitions (so can I, and a myriad other people with basic
mathematical understanding). He applies the definition. Why don't you
try applying the standard definition, to see what happens? You need to
put your hyperactive intuition system on hold though.

<big snip>

Since you are posting in sci.math, I assume you are interested in
learning mathematics. If you aren't, why /are/ you posting here?

Well, _are_ you interested in learning any mathematics? If you did, you
might plausibly [well, _almost_ plausibly, anyway] be able to turn some
of your existing ideas into mathematical ones, with which you might
even be able to convey to other people. It seems you wish to have an
"infinite induction" axiom, according to which it is the case that for
a "Tlimit" concept, and any function f not in a
"Tony-special-case-list", it is the case that:

Tlimit(f()) = f(Tlimit())

Now we know you have a "special case exclusion", which you phrase as
something about "differences going to 0" - you are going to have to do
a _lot_ more work before that convinces anyone it means other than "Ask
Tony if this function is OK". But even leaving that aside, it's very
hard to see how the definition of Tlimit (of a sequence) is going to be
anything less than the entire description of the sequence. Informally,
you want more or less any property of the sequence to leave a _trace_
("infinitesimal corners", stuff like that) in the Tlimit; I don't see
how the Tlimit can ever capture any sort of generalisation at all.

Brian Chandler
http://imaginatorium.org

.

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