Arithmetization of the field of reals with exponentiation extended abstract

(1) Shepherdson proved that a discrete unitary commutative semi-ring A satisfies IE0 (induction scheme restricted to quantifier free formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory T in the language of rings plus exponentiation such that the models (A, expA) of T are all integral parts A of models M of T with A closed under expM and expA = expM A. Namely T = EXP, the basic theory of real exponential fields; T = EXP+ the Rolle and the intermediate value properties for all 2-polynomials; and T = Texp, the complete theory of the field of reals with exponentiation. (2) Texp is recursively axiomatizable iff Texp is decidable. Texp implies LE0(x ) (least element principle for open formulas in the language <,+,×,−1, x) but the reciprocal is an open question. Texp satisfies “provable polytime witnessing”: if Texp proves ∀x∃y : |y| < |x|k)R(x, y) (where |y| := log(y) , k < ω and R is an NP relation), then it proves ∀x R(x, f(x)) for some polynomial time function f . (3) We introduce “blunt” axioms for Arithmetics: axioms which do as if every real number was a fraction (or even a dyadic number). The falsity of such a contention in the standard model of the integers does not mean inconsistency; and bluntness has both a heuristic interest and a simplifying effect on many questions – in particular we prove that the blunt version of Texp is a conservative extension of Texp for sentences in ∀∆0(x) (universal quantifications of bounded formulas in the language of rings plus x). Blunt Arithmetics – which can be extended to a much richer language – could become a useful tool in the non standard approach to discrete geometry, to modelization and to approximate computation with reals. Mathematics Subject Classification. 03H15.