Elementary diagrams

Let $\mathcal{M}$ be a nonempty set structure for a set signature $L$. Form $L_M$ by adjoining one distinct fresh constant $c_a$ for every $a\in M$. Use a tagged disjoint copy for the new symbol set (and the canonical tagged inclusion of the old symbols), so no old symbol is identified with a name. This is a set signature by Set signatures and finite syntax strings. Let ${\mathcal{M}}_M$ be the expansion interpreting $c_a$ as $a$ and keeping all old interpretations.

The elementary diagram is

$\mathrm{EDiag}(\mathcal{M})=\{\sigma \in {\mathrm{Sent}}_{L_M}:{\mathcal{M}}_M\models \sigma \}\mathrm{.}$

The set of sentences is a subset of the set of finite words, and the satisfaction relation is a set uniformly definable from the structure by Existence and uniqueness of set satisfaction. Sentence truth is assignment independent by Coincidence for term values and satisfaction; since $M$ is nonempty, fix one assignment, for example the constant assignment at any one $m_0\in M$. Separation therefore gives the displayed set, independently of that assignment. This is a theory in the sense of Theories, models and semantic consequence, containing every true expanded-language sentence, including quantified ones.

For each $a\mathrm{\ne }b$ in $M$, $\mathrm{\neg }(c_a=c_b)$ belongs to the diagram because the two names have distinct interpretations and logical equality is literal equality. For $a=b$, $c_a=c_a$ belongs instead. Naming all elements does not assert that every model of the diagram has only named elements.

Conventions and prerequisites: Elementary embeddings, substructures and chains.