Transitive set

A set $X$ is transitive when every element of an element of $X$ is itself an element of $X$.

A transitive model of set theory is a transitive set or transitive class whose membership relation is the ambient membership relation and which satisfies the specified set-theoretic axioms.

A model of set theory is well-founded when its internally interpreted membership relation is a well-founded relation externally. By the Mostowski collapse theorem, every well-founded extensional set model is isomorphic to a transitive model.

The ordinal height of a model $M$ is its class of internal ordinals. For a transitive set model this is an ordinal $\mathrm{Ord}\cap M$.

Let $M$ be a transitive set model of ZFC. If $\mathrm{Ord}\cap M$ were countable, then every $x\in M$ would be countable: the internal axiom of choice supplies a bijection from $x$ to an ordinal of $M$, which is externally countable. For every $\alpha \in \mathrm{Ord}\cap M$, the internal rank $V_{\alpha }^M$ is an element of $M$ and hence countable. The Axiom schema of replacement inside $M$ gives $M={\bigcup }_{\alpha \in \mathrm{Ord}\cap M}{V}_{\alpha }^M$, a countable union of countable sets. Thus every uncountable transitive set model has uncountably many ordinals.

Assuming $V=L$, every uncountable transitive model of ZFC contains all ambient countable ordinals and witnesses their countability internally. Its ordinal height is at least ${\omega }_1$, and absoluteness of constructible levels puts $L_{{\omega }_1}$ inside it. Every countability witness for a countable ordinal can be chosen in $L_{{\omega }_1}$ by hereditarily countable constructible sets appear below omega-one.