Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 6 / ii / Solution

A sufficient syntactic condition is that $\varphi$ be a sigma-one formula in set theory, written ${\Sigma }_1$: an existential block $\exists z_1\cdots \exists z_m\theta$ with $\theta$ bounded, meaning that each of its quantifiers is of the form $\forall t\in a$ or $\exists t\in a$. A bounded formula is also called ${\Delta }_0$ and is absolute in both directions between transitive classes.

Prove the latter fact by structural induction. Atomic membership and equality have their actual meanings in both classes, and Boolean connectives preserve equivalence of truth. For a bounded quantifier with bound $a\in U_1$, transitivity ensures that every element of $a$ belongs to $U_1$; thus its range is exactly the same in both classes, and induction applies to its matrix. This proves the bounded case. If the smaller class satisfies a ${\Sigma }_1$ formula, take its existential witnesses in $U_1$. They remain available in $U_2$, and bounded absoluteness preserves the matrix. Hence ${\Sigma }_1$ formulas are upward absolute. An arbitrary unbounded universal quantifier has no such witness-preservation argument.