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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 $N$.

Let $N$ be a well-founded model of set theory of the complete theory $T$, and suppose that a Paris model $M\models T$ were ill-founded. Its internal ordinals would then contain an external descending sequenceFor every $n$, choose a first-order formula ${\phi }_n$ that uniquely defines ${\alpha }_n$ in $M$. The sentences asserting that ${\phi }_n$ uniquely defines an ordinal and that the object defined by ${\phi }_{n+1}$ belongs to the object defined by ${\phi }_n$ are true in $M$. Since $T$ is complete, all its models satisfy the same first-order sentences, so the corresponding uniquely defined ordinals in $N$ form an external descending membership sequence. This contradicts the well-foundedness of $N$. Thus Paris models are well-founded when their complete theory has a well-founded model proves that every Paris model of $T$ is well-founded.