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Shepherdson proved that a discrete unitary commutative semi-ring
satisfies
(induction scheme restricted to quantifier
free formulas) iff 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 range over axiom systems for ordered fields with
exponentiation; for three values of we provide a theory
in the language of rings plus exponentiation...
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