How to make infinitesimals

Let m be a finitely additive measure on the set N of positive integers such that:

  1. For all subsets A of N, m(A) is defined and is 0 or 1.
  2. m(N) = 1, and m(A) = 0 for all finite A.

(The existence of m requires the Axiom of Choice.)

Now, let S be the set of all sequences {a_n} of real numbers.

Define an equivalence relation E by {a_n} E {b_n} iff m{n: a_n = b_n} = 1.

Now, define *R = S/E.

Writing <a_n> as the equivalence class of {a_n}, define addition, muliplication and ordering in *R by

<a_n> + <b_n> = <a_n + b_n>

<a_n><b_n> = <(a_n)(b_n)>

<a_n> < <b_n> iff m{n: a_n < b_n} = 1

(This can easily be shown to be well-defined.)

Identify a real number b with the equivalence class <b,b,b,b,.....>

Define x in *R to be infinitesimal iff -a < x < a for all positive real numbers a.

Then <1/n> = <1, 1/2, 1/3, 1/4, 1/5, ....> is infinitesimal. So are 0 and <1/n^2>.

0 < <1/n^2> < <1/n>

This page is adapted from Lindstrom's paper, An invitation to NSA (in NSA and its applications, CUP 1988).

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