The Tarski-Seidenberg Principle as a Transfer Tool

Abstract

We show in the first section that a semi-algebraic set defined over a real closed field R can be naturally extended to any real closed field K containing R. We then explore further the Tarski-Seidenberg principle. It has already been used in two different ways: to prove that the projection of a semi-algebraic set is semi-algebraic (in Chap. 2), and to obtain the Artin-Lang homomorphism theorem (in Chap. 4). Here we study some additional applications of the Tarski-Seidenberg principle. We use its full strength in the last section in order to establish certain properties of the extension of semi-algebraic sets to a larger real closed field, and to show that semi-algebraic functions can also be extended. The possibility of transfer given by the Tarski-Seidenberg principle will be used in subsequent chapters.