A Robinson Called Abraham

Speranza

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).