An Application of a Reflection Principle Zofia Adamowicz Leszek Aleksander Kolodziejczyk Pawe lZbierski December 1st, 2003 Abstract We define a recursive theory which axomatizes a class of models of I Δ 0 +Ω 3 + ¬exp all of which share two features: firstly, the set of Δ 0 definable elements of the model is majorized by the set of elements definable by Δ 0 formulae of fixed complexity; secondly, Σ 1 truth about the model is recursively reducible to the set of true Σ 1 formulae of fixed complexity. In the present paper, we define a consistent recursive theory T , implying I ∆ 0 and inconsistent with I ∆ 0 + exp, which has the following two properties: 1) in every model M | = T elements definable by ∆ 0 formulae of fixed quantifier complexity are cofinal among all ∆ 0 definable elements; 2) for every model M | = T , the set of Σ 1 sentences true in M is recursi- vely reducible to the set of true Σ 1 sentences whose ∆ 0 part has fixed quantifier complexity. Thus, T axiomatizes to some extent the phenomenon of the cofinality of elements definable by ∆ 0 formulae with fixed complexity among all ∆ 0 definable elements, and of the reducibility of the set of true Σ 1 sentences to the set of true Σ 1 sentences whose complexity is fixed. From the logical point of view, the idea behind the construction of T seems to be interesting in itself. The axioms of T reduce the validity of aΠ 1 sentence ψ to the validity a sentence expressing (roughly) a form of “consistency” of ψ. To show the consistency of T , we have to be able to build a model in which all “consistent” Π 1 sentences are true. We construct such a model by iterating the following procedure: given a model M satisfying the “consistency” of the Π 1 sentence ψ 0 , we build another 1