On the solvability of the extended ∀∃ ∧ ∃∀⋆ — Ackermann class with identity

Abstract

Dreben and Goldfarb define the class 5.7 to be the class of all schemata of the form

$$\forall y\exists xPy \wedge \forall y_l \forall y_2 \cdots \forall y_n G$$

where G is quantifier-free and contains, apart from the dyadic predicate letter P, only monadic predicate letters. Dreben and Goldfarb prove the docility of the class 5.7, but the solvability of this class is left open (see Dreben and Goldfarb(1979) p. 136 and p.264). We shall give an outline of a proof of the solvability of the class 5.7. In fact, we shall give an outline of a proof of the solvability of an extension of this class.