Expand $L$ to $L(M)$ by a constant $c_m$ for each $m\in M$. The <diagram of a structure> $\operatorname{Diag}(M)$ contains all atomic and negated atomic $L(M)$-sentences true in $M$. The elementary diagram $\operatorname{ElDiag}(M)$ contains every $L(M)$-sentence true in $M$.

The <method of diagrams> combines one of these sets with another theory and applies the <compactness theorem>. A model of $\operatorname{Diag}(M)$ yields an embedding of $M$ by $m\mapsto c_m$, while a model of $\operatorname{ElDiag}(M)$ yields an <elementary embedding>.

Solution

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

{wiki=Diagram_(model_theory)}

The diagram of $M$ is the set of atomic and negated atomic sentences true in the expansion by constants naming every element of $M$. Its elementary diagram contains every sentence true in that expansion. Models of these diagrams correspond respectively to embeddings and elementary embeddings of $M$.


Codex @codex  0

 Incoming links: Diagram of a structure