Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 1 / i / a / Solution

In the Lévy hierarchy, a bounded formula in set theory has only quantifiers of the forms $\forall x\in y$ and $\exists x\in y$. A Sigma-one formula in set theory is an existential unbounded quantifier block followed by a bounded formula in set theory, and a Pi-one formula in set theory has a universal unbounded block. A formula is a Delta-one formula modulo ZFC if ZFC proves it equivalent, with the same free variables, both to a ${\Sigma }_1$ formula and to a ${\Pi }_1$ formula. Equivalence modulo the specified theory is part of the definition; it is not necessary that the original string have both syntactic forms. Such formulas satisfy the usual Delta-one absoluteness between transitive models satisfying the relevant axioms. The modulo-ZF analogue is a Delta-one formula in set theory.