On the Foundations of Nonstandard Mathematics

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