On nonstandard reals and Granular Mathematics

In this paper we discuss two logic techniques of constructing nonstandard models of real number system. The first method is the ultraproduct construction of hyperreals in nonstandard analysis. This method provides infinitesimals as mathematical objects, which can not be formally expressed in the classical account of mathematical analysis. Since every equivalence class of hyperreals modulo infinitesimals can be seen as a granule, such a nonstandard model can be seen as a kind of Granular Mathematics. The second method is the standard compactness argument in mathematical logic. However, both methods cause some troubles: (1) There are some infinitesimals in unexpected forms. (2) The compactness construction of nonstandard models also shows that the standard initial segments of the nonstandard model for infinite sets (which look just like the set of natural numbers or the set of real numbers) are not always legitimate sets, from the set-theoretic point of view, as we usually expect in classical mathematics. We conclude that infinitesimals and nonstandard reals may exist, and we can not claim their nonexistence even using set theory.