Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 1 / i / c / Solution

A first-order formula $\phi (x_1,\dots ,x_n)$ is an absolute formula for membership structures $A$ and $B$ if, for every tuple $\vec{a}\in (A\cap B{)}^n$,The same elements interpret the free variables in both structures; bound variables range over their respective domains. Thus absoluteness requires both directions, rather than only preservation of truth from the smaller structure to the larger one.