Remarks on Tarski's problem concerning (R, +, *, exp)

The chapter presents the elementary theory of the structure (R, +), and the results could be extended to the structure (R, +, exp). Some aspects of on (R, +) are reviewed and its usage is inquired. The decidability of Th(R, +) is a nice result in its own right and quite useful in many theoretical decidability questions but has otherwise not been important in settling open problems. Th(R, +·)= theory of real closed fields is useful in proving properties of real closed fields: in certain cases the only known proof consists of first establishing the property for the field of reals by transcendental methods and then invoking elimination of quantifiers for (R, <,0, 1, +·). This is called Tarski's Principle.