Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 4 / b / ii

With the same parameter $\alpha =({\omega }_1{)}^M$, letThis formula is downward absolute: if the larger transitive model has no such function, then neither can the smaller model, since any witness in the smaller model would remain a witness in the larger one. The model $M$ satisfies ${\phi }_2(\alpha )$, while the generic surjection $\bigcup G$ makes it false in $M[G]$. Therefore it is not upward absolute.