---
tag: ANWT2HEH2Y
kind: statement
role: proposition
name: L has a reflection principle
sources:
  - wikipedia: en:343335@r1372309993
    locator: Constructible universe, section L has a reflection principle
    note: adopted
    mention: wikipedia://en/343335#4UZO
    sections: [Statement]
inventory: [NBWMGF4K2E]
---

## Statement

Proving that the axiom of separation, axiom of replacement, and [[axiom of choice|0HTGB13FE2]] 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 $L$.