L has a reflection principle

Statement

Proving that the axiom of separation, axiom of replacement, and axiom of choice hold in $L$ requires (at least as shown above) the use of a reflection principle for $L$. Here we describe such a principle.

By induction on $n<\omega$, we can use ZF in $V$ to prove that for any ordinal $\alpha$, there is an ordinal $\beta>\alpha$ such that for any sentence $P(z_1,\ldots,z_k)$ with $z_1,\ldots,z_k$ in $L_\beta$ and containing fewer than $n$ symbols (counting a constant symbol for an element of $L_\beta$ as one symbol) we get that $P(z_1,\ldots,z_k)$ holds in $L_\beta$ if and only if it holds in $L$.