Forking is Local

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Theorem

Let $T$ be a complete $\mathcal{L}$-theory.

Let $\mathfrak{C}$ be a monster model for $T$.

Let $A\subseteq B$ be subsets of the universe of $\mathfrak{C}$.

Let $\pi(\bar x)$ be an $n$-type over $B$.


$\pi$ forks over $A$ if and only if a finite subset of $\pi$ forks over $A$.


Proof

The proof is straightforward using the definition of forking and the fact that proofs in first-order logic are finite.


Suppose $\pi$ forks over $A$.

By definition, $\pi$ implies a disjunction of formulas which each divide over $A$.
Since proofs are finite, this means that there is a finite subset of $\pi$ which implies this disjunction, completing this direction of the proof.


Suppose a finite subset of $\pi$ forks over $A$.

By definition, the finite subset implies a disjunction of formulas which each divide over $A$.
But then $\pi$ clearly implies this disjunction as well, completing this direction of the proof.


$\blacksquare$