Alfred Tarski's elimination theory for real closed fields

Abstract

Tarski made a fundamental contribution to our understanding of R, perhaps mathematics’ most basic structure. His theorem is the following.To any formula ϕ(X1, …, Xm) in the vocabulary {0, 1, +, ·, <} one can effectively associate two objects: (i) a quantifier free formula (X1, …, Xm) in(1) the same vocabulary, and (ii) a proofof the equivalence ϕ ↔ that uses only the axioms for real closed fields. (Reminder: real closed fields are ordered fields with the intermediate value property for polynomials.)Everything in (1) has turned out to be crucial: that arbitrary formulas are considered rather than just sentences, that the equivalence ϕ ↔ holds in all real closed fields rather than only in R; even the effectiveness of the passage from ϕ to has found good theoretical uses besides firing the imagination.We begin this survey with some history in §1. In §2 we discuss three other influential proofs of Tarski's theorem, and in §3 we consider some of the remarkable and totally unforeseen ways in which Tarski's theorem functions nowadays in mathematics, logic and computer science.I thank Ward Henson, and in particular Wilfrid Hodges without whose constant prodding and logistic support this article would not have been written.