## Arithmetization of the field of reals with exponentiation extended abstract

CNRS and Paris 7 University, France; bougatas@logique.jussieu.fr

**(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
in the language of rings plus exponentiation such that the
models (*A*, exp_{A}) of 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*is recursively axiomatizable iff

_{exp}*T*is decidable.

_{exp}*T*implies

_{exp}*LE*(least element principle for open formulas in the language <,+,x,-1,

_{0}(x^{y})*x*) but the reciprocal is an open question.

^{y}*T*satisfies “provable polytime witnessing”: if

_{exp}*T*proves ∀

_{exp}*x∃y : |y| < |x|*(where log(

^{k})R(x,y)*y*),

*k < ω*and

*R*is an NP relation), then it proves ∀

*x R(x,ƒ(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*is a conservative extension of

_{exp}*T*for sentences in ∀Δ

_{exp}_{0}(

*x*) (universal quantifications of bounded formulas in the language of rings plus

^{y}*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.

^{y}Mathematics Subject Classification: 03H15

Key words: Computation with reals / exponentiation / model theory / o-minimality

*© EDP Sciences, 2007*