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(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(More)
We show that the arithmetical theory T 0 2 + ˆ Σ b 1-IN D |x| 5 , formalized in the language of Buss, i.e. with x/2 but without the M SP function x/2 y , does not prove that every nontrivial divisor of a power of 2 is even. It follows that this theory proves neither N P = coN P nor S 0 2. Some arithmetical theories are not merely weak but very weak, in the(More)
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