Model of a first-order theory

A structure $M$ is a model of $T$ when $M\models \sigma$ for every sentence $\sigma \in T$.

An element $a$ of a structure $M$ is definable without parameters when some first-order formula $\phi (x)$ uniquely identifies it:

Every element of a pointwise definable model is uniquely specified by a formula with no added parameters. Finite tuples are then uniquely specified by conjunctions of their defining formulas. For a complete theory these formulas isolate the tuple types, so such a model is an atomic model. The converse fails: a countable dense linear order without endpoints with a named point is atomic, while its other points are moved by structure automorphisms fixing that point.

A Paris model of a complete extension $T$ of ZFC is a set model $M\models T$ in which every internal ordinal is a definable element of a first-order structure without parameters.

Let $N$ be a well-founded model of set theory of a complete theory $T\supseteq \mathsf{ZFC}$, and let $M$ be a Paris model of $T$. If the internal ordinals of $M$ admitted an external descending sequence ${\alpha }_0{\ni }^M{\alpha }_1{\ni }^M\cdots$, choose formulas ${\phi }_n$ defining them. Completeness makes $M$ and $N$ satisfy the same sentences, so the unique ordinals defined by the ${\phi }_n$ in $N$ would form an externally descending membership sequence. This contradicts well-foundedness of $N$.

Under the hypothesis of Paris models are well-founded when their complete theory has a well-founded model, a Paris model $M$ is well-founded. Every automorphism of $M$ is then the identity by external epsilon induction: once all members of $x$ are fixed, extensionality gives $\pi (x)=\{\pi (y):y{\in }^{M}x\}=x$. Hence $M$ is a rigid first-order structure.