= Paris models are well-founded when their complete theory has a well-founded model
{c}
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 $\varphi_n$ defining them. Completeness makes $M$ and $N$ satisfy the same sentences, so the unique ordinals defined by the $\varphi_n$ in $N$ would form an externally descending membership sequence. This contradicts well-foundedness of $N$.
Back to article page
