ON THE FOUNDATIONS OF NONSTANDARD MATHEMATICS

摘要: In this paper we survey various set-theoretic approaches that have been proposed over the last thirty years as foundational frameworks for use of nonstandard methods in mathematics. Introduction. Since early developments calculus, infinitely small and large numbers object constant interest great controversy history fact, while on one hand fundamental results differential integral calculus were first obtained by reasoning informally with infinitesimal quantities, it was easily seen their without restrictions led to contradictions. For instance, Leibnitz constantly used infinitesimals his studies (the notation dx is due him), also formulated so-called transfer principle, stating those laws hold about real extended number system including infinitesimals. Unfortunately, neither he nor followers able give a formal justification principle. Eventually, order provide rigorous logical framework treatment line, banished from replaced eδ-method during second half nineteenth century. 1 A correct had wait new field mathematics, namely mathematical logic and, particular, its branch called model theory. basic fact theory every infinite structure has models, i.e. non-isomorphic structures which satisfy same elementary properties. other words, there are different but equivalent structures, sense they cannot be distinguished means properties satisfy. slogan, could say mathematics “words not enough describe reality”. 1An interesting review can found Robinson’s book [R2], chapter X.