Re: Differentiability at a point: understanding starting from set theory; request for help.
- From: rem642b@xxxxxxxxx (Robert Maas, see http://tinyurl.com/uh3t)
- Date: Sat, 11 Jun 2005 10:22:46 -0700
> From: "A. Boom" <aboom@xxxxxxxxxx>
> I've not learned yet what "p-adic" functions are.
I don't what you mean by p-adic functions. All I know about are p-adic
norms which then produce p-adic metrics (and "p-adic numbers" which are
elements of such a normed linear space or metric space respectively, a
slight misnomer in the same way that "real number" as member of
real-metric space is a slight misnomer). Are you mixing up the terms,
or is there something called p-adic functions which you saw somewhere
else and mentionned here? I did a Google search and see a reference to
p-adic automorphic functions, which are beyond my understanding, and a
couple other contexts that aren't available in ASCII so I can't see the
text at all.
> > Now in elementary calculus, which deals with the reals, you've already
> > completed the rationals using the real metric derivived from the real
> > norm, so you're stuck taking limits and derivatives using that
> > metric/norm instead of one of the p-adic metrics/norms.
> Ah! That is some understanding that I was after! As I understand it, I
> would need to go further back in the definitions outside of calculus, to
> the very concept of real numbers, in order to apply a meaningful change
> in the metric.
That's correct. To use a different metric, you need to back away from
the normed linear field (or metric space) we call "real numbers" all
the way back to rational numbers, and then build forward from there
using a different metric, arriving at one of the p-adic fields/spaces.
> > Another example: If the value of the norm is always an integer, and
> > another property is satisfied (can you guess what it is?), then the
> > topology is exactly the integers under the real norm.
> That the value of the norms is ordered?
I don't understand that question. It doesn't seem to make any sense.
The numbers upon which the norm is applied are differences between the
numbers/points we call integers. If you don't know what the
numbers/points are a priori, but somebody tells you that the norm of
the difference is always an integer, and tells you this other propery
is also globally satisfied among the set of points under consideration,
you can derive that the structure you have defined is isomorphic to the
integers (as points in the space).
> Well, is the flat-triangle condition that the abstract space has a
> constant metric at all points and that it is nothing but the normal
> Euclidean one?
Nope. By "flat triangle" condition, which I invented for this exercise,
I mean that all triangles are "flat", i.e. one side (length) of the
triangle is exactly equal to the sum of the other two sides (lengths),
no actual triangles as we know them in geometry, all triangles are
degenerate sets of colinear points per distance measurement.
> > Define B between A and C if d(A,B) + d(B,C) = d(A,C).
> > If for all points A,B,C, at least one of the points is between the
> > other two, then this betweenness property can directly generate a total
> > ordering, as soon as you've made one arbitrary binary decision.
> > (If you haven't guessed, betweenness and flat-triangle are same thing.)
> > Can you see the trivial definition of the total ordering?
> Yes, I see how that definition if modified to an "iff" would work. It
> seems to the normal intuitive understanding of what it means for
> something to be between two others. All elements could be put into a
> list using the betweenness property,
Correct, but...
> however, in order for it to be an
> ordering the first (or last) element would need to be specified.
Not correct. There's no such thing as a first or last element among the
integers. Still the integers are totally ordered, you agree?
> That
> would, in my mind, rotate the list around to whichever way you wanted.
Correct. That's why you must make one arbitrary binary decision before
you have defined the ordering. Without that decision yet made, you have
two possible orderings, forward and backward. But if you pick any two
distinct elements x and y and arbitrary decide that x comes before y,
then you've eliminated one of the two possible orderings, so you've now
precisely defined the ordering on this set, which is order-isomorphic
to the integers.
The integers themselves have one additional arbitrary decision
included: The origin (the zero point). There are many
order-isomorphisms between the set defined above and the integers, or
between two copies of the integers, simply by picking different origins
in one or another set. But if you arbitrariy pick one element of the
ordered set and define it to be zero, then there's exactly one
order-and-zero-preserving isomorphism between our set and the integers,
and Peano's postulates apply to the portion of our set to the right of
that zero point, and all of arithmetic and analysis follows.
.
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