This book is concerned with an attempt to introduce the infinitesimals and the
other ‘nonstandard’ numbers in a naive, simpleminded way. Nevertheless, the
resulting theory is hoped to be mathematically sound, and to be complete within obvious limits. Very likely, however, even if ‘nonstandard analysis’ is presented naively, we cannot do without the axiom of choice (there is a restricted version
of nonstandard analysis, less elegant and less powerful, that does not need it). This is a pity, because this axiom is not obvious to every mathematician, and is even rejected by constructivistic mathematicians, which is not unreasonable as it
does not tell us how the relevant choice could be made (except in simple cases, but then the axiom is not needed). |