The new introduction of infinitesimals in the non-standard analysis of Abraham Robinson (since Jones was quoting him)
outdates a remark of Russell.
Abraham Robinson develops his theory on the basis of a fertile use of
Cantor's theory of actually infinite sets (transfinite cardinalities).
A number a is called infinitesimal (or
infinitely small) if its absolute value is less than m for all positive numbers m in (being the set of real
numbers).
According to this definition, 0 is infinitesimal.
The fact that the infinitesimal is merely the
correlate of Cantor's transfinite numbers, is apparent in that r (not equal to 0) is infinitesimal if and only
if r to the power of minus 1 (r-1) is infinite (cf. Robinson, 1966:55ff).
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