Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 1 / b / Solution

On $A={\bigcup }_i{A}_i$, interpret a constant at its common value, a function on a tuple by choosing one $A_i$ containing the tuple, and a relation similarly. Total ordering of the indices and compatibility of substructures make these definitions independent of the chosen stage. Every $A_i$ is then a substructure of $A$.

For an elementary chain, induction on formulas provesThe atomic step follows from the induced structure, Boolean steps are immediate, and for an existential formula any witness in the union lies in a later $A_j$ containing the parameters; elementarity between $A_i$ and $A_j$ moves existence back to $A_i$. This is the elementary chain theorem.