Lefschetz Principle (First-Order)

Theorem
Let $\phi$ be a sentence in the language $\mathcal{L}_r = \{0,1,+,-,\cdot\}$ for rings, where $0,1$ are constants and $+,-,\cdot$ are binary functions.

The following are equivalent:


 * 1) $\phi$ is true in every algebraically closed field of characteristic $0$.
 * 2) $\phi$ is true in some algebraically closed field of characteristic $0$.
 * 3) $\phi$ is true in algebraically closed fields of characteristic $p$ for arbitrarily large primes $p$.
 * 4) $\phi$ is true in algebraically closed fields of characteristic $p$ for sufficiently large primes $p$.

Note in particular that since $\C$ is an algebraically closed field of characteristic $0$, these are equivalent to $\phi$ being true in the field $\C$.

Proof
(1)$\iff$ (2):

This follows easily from the theorem that the theory $ACF_p$ of algebraically closed fields of characteristic $p$ is complete, which says that all such fields satisfy the exact same $\mathcal{L}_r$ sentences.

(2)$\implies$(4):

If $\phi$ is true in some such field, then $ACF_0 \models\phi$. By Gödel's Completeness Theorem and the finiteness of proofs, it follows that there is a finite subset $\Delta$ of $ACF_0$ such that $\Delta\models\phi$. Such a $\Delta$ can only make finitely many assertions about the character of its models, so as long as $p$ is selected sufficiently large, an algebraically closed field of characteristic $p$ will satisfy $\phi$.

(4)$\implies$(3):

This case is trivial, since all sufficiently large $p$ work, we can find arbitrarily large $p$ that work.

(3)$\implies$(2):

We prove this by contraposition. Suppose there is no algebraically closed field of characteristic $0$ where $\phi$ is true. Then, $\phi$ is false in algebraically closed fields of characteristic $0$, and since $ACF_0$ is complete, this means that $ACF_0 \models \neg\phi$. Similarly to the case of (2)$\implies$(4), there must then be a finite subset $\Delta$ of $ACF_0$ such that $\Delta\models\neg\phi$. But then, for all sufficiently large $p$, we have that $\phi$ is false in the algebraically closed fields of characteristic $p$. Hence, it cannot be true for arbitrarily large $p$.