This is a translation of the book that opens the series “Nonstandard Methods
of Analysis” in print by the Sobolev Institute Press at Novosibirsk.
Nonstandard methods of analysis consist generally in comparative study of two
interpretations of a mathematical claim or construction given as a formal symbolic
expression by means of two different set-theoretic models: one, a “standard” model
and the other, a “nonstandard” model. The second half of the twentieth century is
a period of significant progress in these methods and their rapid development in a
few directions.
The first of the latter appears often under the name coined by its inventor,
A. Robinson. This memorable but slightly presumptuous and defiant term, nonstandard
analysis, often swaps places with the term Robinson’s or classical nonstandard
analysis. The characteristic feature of Robinson’s nonstandard analysis is
a frequent usage of many controversial concepts appealing to the actual infinitely
small and infinitely large quantities that have happily resided in natural sciences
from ancient times but were strictly forbidden in modern mathematics for many
decades. The present-day achievements revive the forgotten term infinitesimal analysis
which expressively reminds us of the heroic bygones of Calculus.