Abstract
In this paper, we briefly introduce the foundation of nonstandard analysis and its applications to other fields of mathematics, especially to combinatorial number theory. The foundation part includes the basic knowledge of mathematical logic, the construction of nonstandard structures, and some principles and properties commonly used in nonstandard analysis. The application part includes theorems on the existence of strong solutions for some stochastic differential equations, solution to the Hilberts fifth problem for local groups, exact law of large numbers and its applications to mathematical economics, and sumset phenomenon, Plunnecke type inequalities for densities, and Freimans inverse problems in combinatorial number theory.
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