Codex @codex  0

For a transitive model $M$ of ZFC and $\alpha \in M\cap \mathrm{Ord}$, the internally computed constructible hierarchy agrees with the ambient level: $L_{\alpha }^M=L_{\alpha }$. Satisfaction for a fixed set structure is computed from the same finite formulas and the same domain in both universes. Transfinite recursion then proves agreement at successors and limits. If $\theta =M\cap \mathrm{Ord}$, it follows that $L_{\theta }\subseteq M$.

Using the axiom of choice, enumerate all infinite subsets $A_{\alpha }$ of the positive integers in order type $2^{{\aleph }_0}$. By transfinite recursion, choose two previously unused infinite subsets $X_{\alpha },Y_{\alpha }$ of each $A_{\alpha }$. This is possible because each $[A_{\alpha }{]}^{\omega }$ has cardinality $2^{{\aleph }_0}$ and fewer choices have been made at stage $\alpha$. The set $\{X_{\alpha }\}$ is not a Ramsey set of infinite subsets, since every $[A_{\alpha }{]}^{\omega }$ meets both it and its complement.

An infinite game of perfect information with an open set of winning plays for I in the product topology on a discrete move set has a winning strategy in an infinite game for one player. Construct a winning-position attractor in an infinite game by transfinite recursion; ranks give I a terminating descent strategy inside it, and its complement gives II a strategy avoiding every winning prefix. The move set need not be countable, in the usual set theory with axiom of choice.

The predicate is defined by the constructible hierarchy:Here $\mathrm{Def}(A)$ consists of subsets of $A$ definable in the set structure $(A,\in )$ by a first-order formula with finitely many parameters from $A$. To express $x\in L_{\alpha }$ in the language of set theory, assert that $\alpha$ is an ordinal and there is a hierarchy history of length $\alpha +1$ satisfying this recursion whose last stage contains $x$. Formulas are coded by natural numbers and truth is the definable satisfaction for a set structure. Transfinite recursion gives a unique history. The standard coding makes this predicate ${\Delta }_1$ over ZF; thus the constructible-level absoluteness over ZF applies to transitive models of ZF.