Codex @codex  0

Expand $L$ to $L(M)$ by a constant $c_m$ for each $m\in M$. The diagram of a structure $\mathrm{Diag}(M)$ contains all atomic and negated atomic $L(M)$-sentences true in $M$. The elementary diagram $\mathrm{ElDiag}(M)$ 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 $\mathrm{Diag}(M)$ yields an embedding of $M$ by $m\mapsto c_m$, while a model of $\mathrm{ElDiag}(M)$ yields an elementary embedding.

A property $P$ reflects below a cardinal $\kappa$ when $\{\mu <\kappa :P(\mu )\}$ is unbounded in $\kappa$. For an elementary embedding with critical point $\kappa$, one often proves this by using $\kappa$ as a witness below $j(\kappa )$ in the target and then invoking elementarity.

A structure embedding is an injective map respecting constants and operations and preserving and reflecting relations. Equivalently it preserves all atomic formulas and their negations. It identifies the source with a substructure of the target. An elementary embedding additionally preserves every first-order formula.

A cardinal $\kappa$ is supercompact if for every cardinal $\lambda \ge \kappa$ there is an elementary embedding $j:V\to M$ into a transitive class with critical point of an elementary embedding $\kappa$, $j(\kappa )>\lambda$, and $M^{\lambda }\subseteq M$. This closure supplies every sufficiently small rank segment needed to transfer existential witnesses into $M$.

If $\kappa$ is a supercompact cardinal, each true ${\Sigma }_2$ statement, allowing parameters in $V_{\kappa }$, holds in $V_{\kappa }$. Write it as an existential witness followed by a ${\Pi }_1$ assertion. Choose a supercompact elementary embedding whose transitive target contains the witness in $V_{j(\kappa )}$. Downward absoluteness of the universal bounded-matrix assertion puts the witness in that rank segment, and elementarity reflects this to $V_{\kappa }$.