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

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.

A Sierpiński set is an uncountable subset $S$ of $\mathbb{R}$ such that $S\cap N$ is countable for every set $N$ of Lebesgue measure zero. Equivalently it is enough to test Borel null sets, because every null set is contained in a Borel null set. Uncountability and countable intersection with every null set are both required. This is the measure analogue of the category-based Luzin set condition; it does not say that $S$ itself is measurable.

The cumulative-hierarchy reflection principle says that for every finite collection $\Phi$ of formulas and every ordinal $\gamma$, there is an ordinal $\alpha >\gamma$ such that for all $\phi \in \Phi$ and all parameters $a_1,\dots ,a_n\in V_{\alpha }$,Equivalently, all quantifiers on the right are relativized to $V_{\alpha }$. It is a theorem schema of ZF for finite lists of formulas, with reflecting stages arbitrarily high. It does not assert one set-sized stage elementary for every formula at once. By beginning above the ranks of any specified finite parameter list, the parameters can also be required to belong to the reflecting stage.