Codex @codex  0

For a transitive set $X$, the relative constructible universe $L(X)$ is the union of the relative constructible hierarchy, starting at $L_0(X)=X$, applying the definable power set at successors and taking unions at limit ordinals. It is the smallest inner model of ZF containing $X$ and all its members. For an arbitrary set $A$, take the base $X$ to be the transitive closure of $\{A\}$, so that $A\in X$. Axiom of choice need not hold for an arbitrary base; the construction uses initial parameters rather than a predicate relativization.

The satisfaction relation says that the finite code $e$ of a first-order formula is true in a set-sized first-order structure $A$ at assignment $\vec{a}$. It is defined uniformly in set theory by recursion on the finite syntax tree: evaluate atomic relations, Boolean operations and quantifiers over the given domain. A code includes only finitely many subformulas, and their truth relations are sets. This does not define a truth predicate for the ambient universe, whose domain is not a set. Finite relation closure for set-theoretic coding gives an explicit closure criterion ensuring that these truth relations are computed correctly in a transitive set.

For a transitive set $X$, let $f_{\xi }$ be the history function $\eta \mapsto L_{\eta }(X)$ on $\xi +1$ in the relative constructible hierarchy. By transfinite induction, $f_{\xi }\in L_{\xi +k}(X)$ for some finite $k$, possibly depending on $\xi$. The base and successor steps follow by forming finite ordered pairs and adjoining the next entry, using $L_{\xi +1}(X)\in L_{\xi +2}(X)$. These operations require finitely many subsequent definable power sets.

At a nonzero limit ordinal $\lambda$, the inductive bounds put every earlier $f_{\eta }$ in $L_{\lambda }(X)$, since $\eta +k<\lambda$. This level satisfies finite relation closure for set-theoretic coding, independently of the availability of history functions. Consequently its coded relative constructible stage predicate computes the endpoints correctly. Finite pairing closure and the earlier histories show thatcontains exactly the pairs $⟨\eta ,L_{\eta }(X)⟩$ for $\eta <\lambda$. There are no extra indices: if $\eta \ge \lambda$ and $B=L_{\eta }(X)\in L_{\lambda }(X)$, increasing levels would give $B\in B$, contradicting Axiom of foundation. Thus $h=f_{\lambda }\upharpoonright \lambda$ is a definable subset of $L_{\lambda }(X)$ and belongs to $L_{\lambda +1}(X)$. Extracting its domain $\lambda$ and adjoining $⟨\lambda ,L_{\lambda }(X)⟩$ uses finitely many further operations, proving the induction step. Every nonzero limit $\alpha$ therefore contains all $f_{\xi }$ with $\xi <\alpha$, as required by relative constructible level recognition.

If $X$ has a well-founded extensional inherited membership relation and $T\subseteq X$ is a transitive set, its Mostowski collapse theorem map $\pi$ fixes every member of $T$. The recursion $\pi (x)=\{\pi (y):y\in x\cap X\}$ and well-founded relation induction give this pointwise. If also $T\in X$, then $\pi (T)=T$.

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.