Arithmetization of the field of reals with exponentiation extended abstract

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 $_{\llcorner} T _{\lrcorner}$ in the language of rings plus exponentiation such that the models (A, exp_A) of $_{\llcorner} T _{\lrcorner}$ 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)$_{\llcorner}$T_exp$_{\lrcorner}$ is recursively axiomatizable iff T_exp is decidable. $_{\llcorner}$T_exp$_{\lrcorner}$ implies LE₀(x^y) (least element principle for open formulas in the language <,+,x,-1,x^y) but the reciprocal is an open question. $_{\llcorner}$T_exp$_{\lrcorner}$ satisfies “provable polytime witnessing”: if $_{\llcorner} $T_exp$_{\lrcorner}$ proves ∀x∃y : |y| < |x|^k)R(x,y) (where $|y|:=_{\llcorner}$log(y)$_{\lrcorner}$, 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 $_{\llcorner} $T_exp$_{\lrcorner}$ is a conservative extension of $_{\llcorner} $T_exp$_{\lrcorner}$ for sentences in ∀Δ₀(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.