The Stacks project

3.8 Reflection principle

Some of this material is in the chapter of [Kunen] called “Easy consistency proofs”.

Let $\phi (x_1, \ldots , x_ n)$ be a formula of set theory. Let us use the convention that this notation implies that all the free variables in $\phi $ occur among $x_1, \ldots , x_ n$. Let $M$ be a set. The formula $\phi ^ M(x_1, \ldots , x_ n)$ is the formula obtained from $\phi (x_1, \ldots , x_ n)$ by replacing all the $\forall x$ and $\exists x$ by $\forall x\in M$ and $\exists x\in M$, respectively. So the formula $\phi (x_1, x_2) = \exists x (x\in x_1 \wedge x\in x_2)$ is turned into $\phi ^ M(x_1, x_2) = \exists x \in M (x\in x_1 \wedge x\in x_2)$. The formula $\phi ^ M$ is called the relativization of $\phi $ to $M$.

We view this theorem as saying the following: Given any $x_1, \ldots , x_ n \in M$ the formulas hold with the bound variables ranging through all sets if and only if they hold for the bound variables ranging through elements of $V_\alpha $. This theorem is a meta-theorem because it deals with the formulas of set theory directly. It actually says that given the finite list of formulas $\phi _1, \ldots , \phi _ m$ with at most free variables $x_1, \ldots , x_ n$ the sentence

is provable in ZFC. In other words, whenever we actually write down a finite list of formulas $\phi _ i$, we get a theorem.

It is somewhat hard to use this theorem in “ordinary mathematics” since the meaning of the formulas $\phi _ i^ M(x_1, \ldots , x_ n)$ is not so clear! Instead, we will use the idea of the proof of the reflection principle to prove the existence results we need directly.