Infinitesimal analysis. Part I. (Q2728730)

The purpose of this book is to make new ideas in infinitesimal analysis related to Robinson's nonstandard analysis and Nelson's theory of internal sets more accessible. To this end, the authors start with presenting the semantic qualitative views of standard and nonstandard objects as well as the relevant apparatus at the ``naive'' level of rigor which is absolutely sufficient for effective applications without appealing to any logical formalism. They then give concise reference material pertaining to the modern axiomatic expositions of infinitesimal analysis within the classical Cantorian doctrine. They have found it appropriate to allot plenty of room to the ideological and historical facets of the topic, which has determined the plan and style of exposition. NEWLINENEWLINENEWLINEThe contents is as follows. Chapter 1 ``Excursions in the history of calculus'' (pp. 1-11) contains the historical signposts alongside the qualitative motivation of the principles of infinitesimal analysis. Chapter 2 ``Naive foundations of infinitesimal analysis'' (pp. 12-43) contains a discussion of simplest implications of the principles of infinitesimal analysis for differential and integral calculus. This lays the ``naive'' foundation of infinitesimal analysis. Chapter 3 ``Set-theoretic formalism of infinitesimal analysis'' (pp. 44-115) gives formal details of the corresponding apparatus of nonstandard set theory. Chapter 4 ``Monads in general topology'' (pp. 116-165) and Chapter 5 ``Infinitesimals and subdifferentials'' (166-274) set forth the infinitesimal methods of general topology and subdifferential calculus. The list of references (275-309) contains 531 items.NEWLINENEWLINENEWLINEFor Part II, see Zbl 1013.46004 below.