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1 A CONSISTENCY PROOF FOR ELEMENTARY ALGEBRA AND GEOMETRY by Harvey M. Friedman Department of Mathematics Ohio State University August 10, 1999 August 23, 1999 ABSTRACT We give a consistency proof within a weak fragment of arithmetic of elementary algebra and geometry. For this purpose, we use EFA (exponential function arithmetic), and various first order theories of algebraically closed fields and real closed fields. We actually prove in EFA that RCF (real closed fields) is consistent and complete. The completeness proof is an adaptation of known constructions. We also obtain a proof in EFA that every quantifier free formula provable in RCF has a quantifier free proof in the theory of real fields and in the theory of ordered fields. As a Corollary, we obtain an interpretation of RCF into EFA in the sense of model theory, as well as interpretations of finitely axiomatized extensions of RCF into EFA. The development can be used to provide fixed length iterated exponential estimates in connection with Hilbert’s seventeenth problem. This application will appear elsewhere. INTRODUCTION to be written. discusses Hilbert’s program briefly. discusses systems of arithemetic briefly. introduces EFA = exponential function arithmetic. reference to Hajek/Pudlak. disclaimer that we make no attempt here to obtain more precise information. 1. PRELIMINARIES ABOUT LOGIC In the free variable predicate calculus with equality, we start with a signature consisting of a set of constant, relation, and function symbols. Formulas are built up in the usual way using the variables v 1 ,v 2 ,..., connectives not, or,
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2 and, implies, iff, =, and the symbols in . There are no quantifiers. The variables are literally the v i , i 1. We use this to define the universal closure of a formula j . This is obtained from j by successively universally quantifying by the free variables of j in reverse order of their subscripts. If j is a sentence then the universal closure of j is j . The logical axioms are 1. All tautologies in the language. The equality axioms are 2. x = x, where x is a variable. 3. x = y implies ( j implies j ’), where x,y are variables, j , j ’ are atomic formulas, and j ’ is the result of replacing some occurrences of x in j by y. The rules of inference are 4. From j and ( j implies y ) derive y . 5. From j derive any term substitution of j in the language. Suppose we are given a set T of proper axioms; i.e., a set of quantifier free formulas in the language. Then a proof in T is a finite sequence of formulas, every one of which is either a logical axiom, an equality axiom, a proper axiom, or follows from previous formulas in the finite sequence by one or more rules of inference. The proof in T is said to be a proof in T of the last formula. We say that the last formula is free variable provable in T.
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