TY - JOUR

T1 - Tarski's problem and pfaffian functions

AU - van den Dries, Lou

N1 - Funding Information:
* Partiah supported by NSF.

PY - 1986/1

Y1 - 1986/1

N2 - 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.

AB - 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.

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U2 - 10.1016/S0049-237X(08)70457-9

DO - 10.1016/S0049-237X(08)70457-9

M3 - Article

AN - SCOPUS:77956968213

SN - 0049-237X

VL - 120

SP - 59

EP - 90

JO - Studies in Logic and the Foundations of Mathematics

JF - Studies in Logic and the Foundations of Mathematics

IS - C

ER -