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Arithmetization of the field of reals with exponentiation extended abstract

Sedki Boughattas, Jean-Pierre Ressayre (2008)

RAIRO - Theoretical Informatics and Applications

 (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 T in the language of rings plus exponentiation such that the ...

Around Podewski's conjecture

Krzysztof Krupiński, Predrag Tanović, Frank O. Wagner (2013)

Fundamenta Mathematicae

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case). We also study minimal groups with a (partial) order, and give a complete classification of...

Constructing ω-stable structures: Computing rank

John T. Baldwin, Kitty Holland (2001)

Fundamenta Mathematicae

This is a sequel to [1]. Here we give careful attention to the difficulties of calculating Morley and U-rank of the infinite rank ω-stable theories constructed by variants of Hrushovski's methods. Sample result: For every k < ω, there is an ω-stable expansion of any algebraically closed field which has Morley rank ω × k. We include a corrected proof of the lemma in [1] establishing that the generic model is ω-saturated in the rank 2 case.

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