Degrees of models

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Abstract
According to Gödel's completeness theorem, every consistent theory1 has a model whose domain is a set of natural numbers. The objects of the model corresponding to the predicate symbols of the theory are then predicates of natural numbers. Kleene [5] p. 398 showed that, if the theory is axiomatizable,2 then the model can be chosen so that these predicates are in both two-quantifier forms, i.e., they can be expressed in both the forms (x)(Ey)R and (Ex)(y)S with R and S recursive. An alternative proof has been given by Hasen jaeger [3].