Tarski's problem and pfaffian functions

This chapter discusses Tarski's problem and presents Pfaffian functions. Tarski's theorem establishes a link between the algebraic–analytic structure of the real field and its logical properties. The Renaissance gave the isomorphism between addition and multiplication: multiplying (positive) reals amounts to adding their logarithms. The (graph of) the exponential function is not semi-algebraic, so the system of semi-algebraic sets is missing a fundamental connection between its two generating operations. The remarkable finiteness properties of semi-algebraic sets are desirable. These finiteness phenomena are consequences of 0-minimality. Pfaffian functions are real analytic functions defined on certain cells called “Pfaffian cells”, and they satisfy differential equations of a certain form. The partial derivatives of a Pfaffian function on an open cell are Pfaffian, and the restriction of a Pfaffian function to a Pfaffian subcell of its domain is Pfaffian.