Reconciling transparency, low <i>Δ</i>0-complexity and axiomatic weakness in undecidability proofs

Abstract In a first-order theory $\varTheta $, the decision problem for class of formulae $\varPhi $ is solvable if there an algorithmic procedure that can assess whether or not existential closure $\varphi ^{\exists }$ belongs to any \in \varPhi $. 1988, Parlamento and Policriti already showed how tailor arguments à la Gödel very weak axiomatic set theory, referring them $\varSigma _{1}$-formulae with $(\forall \exists \forall )_{0}$-matrix, i.e. closures contain just restricted quantifiers forms x y)$ $(\exists are writable in prenex form at most two alternations (the outermost quantifier being ‘$\forall $’). While revisiting their work, we show slightly less theories under which incompleteness recursively axiomatizable extensions holds respect )_{0}$-matrices, namely one alternation quantifiers.