Arithmetization of the field of reals with exponentiation extended abstract

(1) Shepherdson proved that a discrete unitary commutative semi-ring A + satisfies IE 0 (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, exp A ) of [T] are all integral parts A of models M of T with A + closed under exp M and exp A = exp M | A + . Namely T = EXP, the basic theory of real exponential fields; T = EXP+ the Rolle and the intermediate value properties for all 2 x -polynomials; and T = T exp , the complete theory of the field of reals with exponentiation. (2) [T exp ] is recursively axiomatizable iff T exp is decidable. [T exp ] implies LE 0 (x y ) (least element principle for open formulas in the language <, +, x, -1,x y ) but the reciprocal is an open question. [T exp ] satisfies "provable polytime witnessing": if (T exp ] 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 [T exp ] is a conservative extension of [T exp ] for sentences in VΔ 0 (x y ) (universal quantifications of bounded formulas in the language of rings plus x y ). 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.