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Remark 3.9.3. The lemma above can also be proved using the reflection principle. However, one has to be careful. Namely, suppose the sentence $\phi _{scheme}(X)$ expresses the property “$X$ is a scheme”, then what does the formula $\phi _{scheme}^{V_\alpha }(X)$ mean? It is true that the reflection principle says we can find $\alpha $ such that for all $X \in V_\alpha $ we have $\phi _{scheme}(X) \leftrightarrow \phi _{scheme}^{V_\alpha }(X)$ but this is entirely useless. It is only by combining two such statements that something interesting happens. For example suppose $\phi _{red}(X, Y)$ expresses the property “$X$, $Y$ are schemes, and $Y$ is the reduction of $X$” (see Schemes, Definition 25.12.5). Suppose we apply the reflection principle to the pair of formulas $\phi _1(X, Y) = \phi _{red}(X, Y)$, $\phi _2(X) = \exists Y, \phi _1(X, Y)$. Then it is easy to see that any $\alpha $ produced by the reflection principle has the property that given $X \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ the reduction of $X$ is also an object of $\mathit{Sch}_\alpha $ (left as an exercise).


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