We do have math capable of handling the concept of the smallest number
after zero. Its a particular

application of

calculus called

infinitesimal calculus. This
method was invented by Leibniz in the mid 1670s, and published in 1684 -
nine years before the earliest account of Newton's method.

Leibniz developed his calculus in order to find
methods by which discrete *infinitesimal* quantities could be summed
to calculate the area of a larger whole. This probably came from his
metaphysical work on monads. Newton was working with infinitesimal
changes of force and motion with respect to time.

In order to work with the smallest possible number, one must work with
infinitesimals, around which the model of calculus has been developed.

To work with infinitesimals, it necessary to use the
hyperreal number system. This is much closer to the math that
Newton and Leibniz did. In 1960, Abraham Robinson established the
framework for non-standard analysis.

First off, some definitions.

- Internal - objects that exist within classical mathematics
- Standard - limited objects within the internal set

There are three axioms added to ZFC theory (the basis for today's
mathematics): Idealization, Transfer, Standardization. These lead
to the three principles of non-standard analysis:

- 1
^{st} principle: If E is an **internal** object which is
defined from **standard**
objects, then E is standard.
- 2
^{nd} principle: All elements of an internal set are standard
iff the set is finite.
- Transfer principle: Let P(x) be an internal expression relative to x.
Then P(x) is true for all x, iff P(x) is true for all standard x.

For any x (standard or not)

- x is
**limited** iff there is a limited integer greater than x.
- x is
**unlimited** iff it is greater than any limited integer.
- x is
**infinitesimal** iff its absolute value is less than 1/n for
any limited integer n
- x is
**perceptible** iff x is not unlimited or infinitesimal
- x is
**infinitely close** to y iff x - y is infinitesimal

A new function is added called the Standard Part that operates on
the hyperreal number set and maps it back into the real numbers.
For 'e' designating Epsilon (an infinitesimal):

SP(1 + e) = 1

SP(e) = 0

It is not possible to take the standard part of an infinite number,
but it is not difficult to take the standard part of the reciprocal.

Yes, this is a bit hazy in my memory. I happily defer to anyone
who can explain non-standard analysis better than I. The point being,
we do have a formal system for dealing with the smallest number greater
0, and it has been around for awhile.

A text book on infinitesimal calculus has been made available under a Creative Commons License when the book went out of print and reverted back to the professor (incidently, the one I took 2nd semester calculus under) who wrote the book. The book is available via PDF at __http://www.math.wisc.edu/~keisler/calc.html__

*ENOUGH*

*The answer is 1 unless you're a freak.*

No more writeups, please.

An editor who is also a

mathematician writes:

I have just killed thirteen (13) write-ups in the above node, because none of them demonstrated sufficient knowledge of non-standard analysis and infinitesimals to make a useful contribution to it. Trust us, we mathematicians do know what we're doing. All your radical ideas have already occurred to us.