Introduction of Nonstandard Methods for Number Theorists

In the past few decades, nonstandard methods, as a branch of mathematical logic, have been successfully applied to obtain new results in additive/combinatorial number theory (cf. [BJ, Ji1, Ji2, Ji3, Ji4, Ji5, Ji6, JK, Le1, Le2, Le3, Le4]). Although the nonstandard techniques in these applications are elementary from a nonstandard analyst’s point of view, it is extremely difficult for a reader who tries to understand these techniques without basic training in mathematical logic. One of the purposes of this article is to provide the nuts and bolts of this training to those who do not have the logic background so that they can understand the proofs in the papers mentioned above and, hopefully, find new applications of the methods to new problems after reading this article. In short, nonstandard methods take advantage of “infinitely large” integers existing in a nonstandard model. A nonstandard model is a proper extension of the standard world (or standard model) and possesses the same “first-order” truths as in the standard world. Staying inside the nonstandard model, one may not recognize that the model is nonstandard due to the fact that both standard and nonstandard model satisfy the same “firstorder” truth. But if one looks at the nonstandard model from the outside, then one can see many “infinitely large” integers in it. By working inside and outside of the nonstandard model alternatively, one can gain insights as well as simplify logical reasoning process in solving some number theoretic problems. From the experiences of this author some mathematicians often have had doubts that nonstandard methods bring significant advantages to the standard world. As a consequence they may not believe that it is worthwhile to