Order-of-magnitude spaces

An order-of-magnitude space, or *om-space*, is a space of geometric
points. Any two points are separated by a distance. Two distances *d* and
*e* are compared by the relation *d* << *e*, meaning ``Distance *d* is
infinitesimal compared to *e*'' or, more loosely, ``Distance *d* is much
smaller than *e*.''

For example, let *R*^{*} be the non-standard real line with infinitesimals.
Let *R*^{*m} be the corresponding *m*-dimensional space. Then we can
let a point of the om-space be a point in *R*^{*m}. The distance between
two points *a*,*b* is the Euclidean distance, which is a non-negative
value in *R*^{*}. The relation *d* << *e* holds for two distances *d*,*e*,
if *d*/*e* is infinitesimal.

The distance operator and the comparator are related
by a number of axioms, specified below. The most interesting of these is
called the *om-triangle inequality*: If *ab* and *bc* are both
much smaller
than *xy*, then *ac* is much smaller than *xy*. This combines the ordinary
triangle inequality ``The distance *ac* is less than or equal to distance
*ab* plus distance *bc*'' together with the rule from order-of-magnitude
algebra, ``If *p* << *r* and *q* << *r* then *p*+*q* << *r*.''

It will simplify the exposition below if, rather than talking about
distances, we talk about orders of magnitude. These are defined
as follows. We say that two distances *d* and *e* have the *same*
order of magnitude if neither *d* << *e* nor *e* << *d*. In *R*^{*}this is the condition that *d*/*e* is finite: neither infinitesimal
nor infinite. (Raiman, 1990 uses the notation
``*d* Co *e*'' for this relation.)
By the rules of the order-of-magnitude calculus, this is an equivalence
relation. Hence we can define *an order of magnitude* to be
an equivalence class of distances under the relation ``same order of
magnitude''. For two points *a*,*b*, we define the function od(*a*,*b*)
to be the order of magnitude of the distance from *a* to *b*.
For two orders of magnitude *p*,*q*, we define *p* << *q* if, for any
representatives *d in p* and *e in q*, *d* << *e*. By the rules
of the order-of-magnitude calculus, if this holds for any representatives,
it holds for all representatives.
The advantage of using orders-of-magnitude and the function ``od'',
rather than distances and the distance function, is that it allows
us to deal with logical equality rather than the
equivalence relation ``same order of magnitude''.

For example, in the non-standard real line, let *d* be a positive
infinitesimal value. Then values such as
{
1,100,2-50*d*+100*d*^{2}...},
are all of the same order of magnitude, *o*1. The values
{
*d*,1.001*d*,3*d*+*e*^{-1/d}...} are of a different order
of magnitude *o*2 << *o*1. The values {
1/*d*,10/*d*+*d*^{5}...}
are of a third order of magnitude *o*3 >> *o*1.

**Definition 1:**
An *order-of-magnitude space (om-space)* *O* consists of:

- A set of points
*P*; - A set of orders of magnitude
*D*; - A distinguished value 0 in
*D*; - A function ``od(
*a*,*b*)'' mapping two points*a*,*b in P*to an order of magnitude; - A relation ``
*d*<<*e*'' over two orders of magnitude*d*,*e in D*

- A.1
- For any orders of magnitude
*d*,*e in D*, exactly one of the following holds:*d*<<*e*,*e*<<*d*,*d*=*e*. - A.2
- For
*d*,*e*,*f in D*, if*d*<<*e*and*e*<<*f*then*d*<<*f*.

(Transitivity. Together with A.1, this means that << is a total ordering on orders of magnitude.) - A.3
- For any
*d in D*, not*d*<< 0.

(0 is the minimal order of magnitude.) - A.4
- For points
*a*,*b in P*, od(*a*,*b*) = 0 if and only if*a*=*b*.

(The function od is positive definite.) - A.5
- For points
*a*,*b in P*, od(*a*,*b*) = od(*b*,*a*).

(The function od is symmetric.) - A.6
- For points
*a*,*b*,*c in P*, and order of magnitude*d in D*,

if od(*a*,*b*) <<*d*and od(*b*,*c*) <<*d*then od(*a*,*c*) <<*d*.

(The om-triangle inequality.) - A.7
- There are infinitely many different orders of magnitude.
- A.8
- For any point
*a*_{1}*in P*and order of magnitude*d in D*, there exists an infinite set*a*_{2},*a*_{3}... such that od(*a*_{i},*a*_{j}) =*d*for all*i*<>*j*.

The example we have given above of an om-space, non-standard Euclidean space, is wild and woolly and hard to conceptualize. Here are two simpler examples of om-spaces:

I. Let *d* be an infinitesimal value. We define a point to be a
polynomial in *d* with integer coefficients, such as
3 + 5*d* - 8 *d*^{5}. We define an order-of-magnitude to
be a power of *d*. We define
*d*^{m} << *d*^{n} if *m* > *n*;
for example,
*d*^{6} << *d*^{4}. We define od(*a*,*b*) to be
the smallest power of *d in a*-*b*. For example,
od(1+*d*^{2}-3*d*^{3}, 1-5*d*^{2}+4*d*^{4}) = *d*^{2}.

II. Let *N* be an infinite value. We define a point to be a
polynomial in *N* with integer coefficients.
We define an order of magnitude to
be a power of *N*. We define
*N*^{p} << *N*^{q} if *p* < *q*;
for example,
*N*^{4} << *N*^{6}. We define od(*a*,*b*) to be
the largest power of *N in a*-*b*. For example,
od(
1+*N*^{2}-3*N*^{3}, 1-5*N*^{2}+4*N*^{4}) = *N*^{4}.

It can be shown that any om-space either contains a subset isomorphic to (I) or a subset isomorphic to (II). (This is just a special case of the general rule that any infinite total ordering contains either an infinite descending chain or an infinite ascending chain.)

We will use the notation ``*d* <<= *e*'' as an abbreviation for
``*d* << *e* or *d* = *e*''.