Re: Two results of set geometry
- From: lwalke3@xxxxxxxxx
- Date: Sat, 06 Oct 2007 22:51:28 -0700
On Oct 5, 10:20 pm, David R Tribble <da...@xxxxxxxxxxx> wrote:
Can I see the point that's halfway between the center
and the right edge? Is that the point at BigUn/2
(or maybe BigUn/4, I'm losing track) that's a finite
distance from the center (because all the points in
my square are a finite distance from the center)?
I forget, is BigUn/2 a finite number or not?
As usual, I've tried to find a way, using hyperreals, to
make sense of all this Big'Un/Lil'Un/iota, etc.
We begin by identifying Big'Un with a particular
hyperreal -- might as well use the equivalence class
containing the identity sequence:
B = {1, 2, 3, 4, 5, 6, ...}
(using B for Big'Un, of course). This answers the
last question immediately -- Big'Un/2 is still an
infinite hyperreal (B/2), as in B/4, as well as B/n for
any finite n. Proceeding, if we were now to let Z*
denote the set of hyperintegers (i.e., the set of
all hyperreals with at least one sequence exclusively
of integers in its equivalence class), then we define P,
the set of "points," as the set of all hyperreals X such
that BX is an element of Z*.
The smallest positive element of P is 1/B. Now
1/B is obviously infinitesimal -- and we'll denote
that infinitesimal as L (for Lil'Un). We observe a few
things about the set P:
-- For every standard real r, there exists a point X
whose shadow (standard part) is r. To construct
such a point, we consider the hyperreal:
X = {floor(r), floor(2r)/2, floor(3r)/3, floor(4r)/4, ...}
-- With a gratuitous choice of ultrafilter, one can make
Q (the set of all standard rationals) a subset of P. One
such ultrafilter is one containing the set of factorials:
{1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...}
It is interesting to see what sort of geometry arises if
we used these "points" instead of the standard reals.
So L is indeed the next point after 0. Of course, there
are still hyperreals between zero and L, but none of
these would be elements of P. I assume this is what
TO means by "first-level," "second-level" infinitesimals.
So we have L/2, L/4, ..., and even L^2 as TO's
"second-level" infinitesimal. Presumably L^3 would be
"third-level" infinitesimal, L^n would be "nth-level"
infinitesimal, and ultimately L^B, which would be a
"Big'Unth-level" infinitesimal.
RF, on the other hand, intends his iota to be nilpotent,
so there are no higher-level infinitesimals.
.
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- Re: Two results of set geometry
- From: David R Tribble
- Re: Two results of set geometry
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