Theorem 3.8.1 (000G)—The Stacks project

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Theorem 3.8.1. Suppose given $\phi _1(x_1, \ldots , x_ n), \ldots , \phi _ m(x_1, \ldots , x_ n)$ a finite collection of formulas of set theory. Let $M_0$ be a set. There exists a set $M$ such that $M_0 \subset M$ and $\forall x_1, \ldots , x_ n \in M$ , we have

$\forall i = 1, \ldots , m, \ \phi _ i^{M}(x_1, \ldots , x_ n) \Leftrightarrow \forall i = 1, \ldots , m, \ \phi _ i(x_1, \ldots , x_ n).$

In fact we may take $M = V_\alpha$ for some limit ordinal $\alpha$ .

Proof. See [Theorem 12.14, Jech] or [Theorem 7.4, Kunen]. $\square$

Comments (2)

Comment #3542 by Laurent Moret-Bailly on August 27, 2018 at 05:17

Is there any reason for stating the theorem in terms of finitely many formulas rather than just one?

Comment #3674 by Johan on October 22, 2018 at 10:23

Going to leave this as is. But, no, I guess not, because you can replace the list of $\phi_i$ by $\phi_1 \wedge \ldots \wedge \phi_m$ .

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