  In 1964 Shepherdson proved the following fact: a discretely ordered semiring
$\mathcal{M}^+$ satisfies $\sf{IOpen}$ (open, or quantifier free, induction)
iff the corresponding ring $\mathcal{M}$ is an integer part of the real closure
of the quotient field of $\mathcal{M}$. In this paper, we consider the open
induction in the expanded languages with the exponentiation and the power
function and try to find a similar criterion for models of these theories.
  For several expansions $\mathsf T$ of the theory of real closed fields we
obtain analogues of Shepherdson's Theorem in the following sense: if an
exponential field $\mathcal R$ is a model of $\mathsf T$ and discretely ordered
ring $\mathcal M$ is an (exponential) integer part of $\mathcal R$, then
$\mathcal M^+$ is a model of the open induction in the expanded language. The
proof of the opposite implications remains an open question, but for some
finite extension of the usual open induction we can prove the following
theorem: for any discretely ordered ring $\mathcal M$ holds $\mathcal M^+$
satisfies this finite extension iff there is an exponential real closed field
with the Bernoulli inequality $\mathcal R$ such that $\mathcal M$ is an
exponential integer part of $\mathcal R$. After that, we consider a nonstandard
o-minimal exponential field, constructed in the paper of van den Dries,
Macintyre, Marker and using it and the results above, construct nonstandard
models of the open induction in the expanded languages and obtain independence
results.

Logic