Model completeness

A theory T is model complete if it satisfies one of the following equivalent conditions:
 * 1) Whenever M ⊆  N is an inclusion of models of T, M is an elementary substructure of N.
 * 2) Every model of T is existentially closed.
 * 3) Every formula $$\phi(x)$$ is equivalent, mod T, to a universal formula.
 * 4) Every formula is equivalent, mod T, to an existential formula.
 * 5) Every universal formula is equivalent, mod T, to an existential formula.
 * 6) Every existential formula is equivalent, mod T, to a universal formula.

See below for a proof that these notions are equivalent.

Examples
Theories with quantifier elimination are model complete. This includes ACF, RCF (in the language with ≤), DLO, and ACVF. Even in the pure field language, RCF is model complete, because x ≤ y admits an existential definition
 * $$ \exists z : y - x = z^2 $$

and a universal definition
 * $$ \forall z, w : w(x - y) \ne 1 \vee z^2 \ne (x - y) $$

Another notable model complete theory, without quantifier elimination, is ACFA.

Proof that the definitions are equivalent
Conditions (3) and (4) are equivalent by De Morgan's Laws: $$\phi(x)$$ is equivalent to a universal formula if and only if $$\neg \phi(x)$$ is equivalent to an existential formula. Conditions (5) and (6) are similarly equivalent.

Conditions (3-4) clearly imply (5-6). Conversely, assume (5-6). We show by induction on n that any formula of the form
 * $$ \forall y_1 \exists y_2 \forall y_3 \cdots Q y_n : \phi(x,y_1,\ldots,y_n)$$

is equivalent to a universal formula, where the yi's are tuples, Q is the appropriate quantifier, and $$\phi$$ is quantifier-free. The base case where n = 1 is trivial.

For n > 1, the inductive hypothesis implies that
 * $$ \neg \exists y_2 \forall y_3 \cdots Q y_n : \phi \equiv \forall y_2 \exists y_3 \cdots \neg \phi$$,

must be equivalent to a universal formula. Consequently,
 * $$ \exists y_2 \forall y_3 \cdots Q y_n : \phi $$

is equivalent to an existential formula. By condition (6), it is equivalent to some universal formula
 * $$ \forall z : \psi(x;y_1;z) $$

Then the original formula is equivalent to
 * $$ \forall y_1 \forall z : \psi(x;y_1;z) $$

which is a universal formula. This completes the inductive proof that any formula in prenex form is equivalent to a universal formula. Any formula can be put in prenex form, so condition (3) holds.

So conditions (3-6) are all equivalent.

It remains to show that (1) implies (2) implies (3-6) implies (1).

Suppose (1) holds, and let us prove (2). We need to prove that whenever M ⊆ N is an inclusion of models of T, then M is existentially closed in M. This means that for every existential formula $$\phi(x)$$ and every tuple a from M, we have
 * $$ M \models \phi(a) \iff N \models \phi(a)$$

Obviously this is a weaker condition than M being an elementary substructure of N, so (1) certainly implies (2).

Now assume (2). Then whenever M ⊆ N is an inclusion of models of T, M is existentially closed in T. So if $$\phi(x)$$ is a formula and c is a tuple from M, then
 * $$ M \models \phi(a) \implies N \models \phi(a)$$

But this is one of the characterizations of universal formulas, so $$\phi(x)$$ must be equivalent (mod T) to a universal formula. Therefore (6) holds.

Finally, assume (3-6). Let M ⊆ N be an inclusion of models. We need to show that M is an elementary substructure of N. Let $$\phi(x)$$ be a formula, and c be a tuple from M. We need to show that
 * $$ M \models \phi(c) \iff N \models \phi(c) \qquad (*)$$

By (3) and (4), we can find existential and universal formulas $$\psi(x)$$ and $$\chi(x)$$, respectively, which are equivalent mod T to $$\phi(x)$$.

Now universal formulas are always preserved downwards, and existential formulas are preserved upwards, so
 * $$ M \models \psi(c) \implies N \models \psi(c) $$
 * $$ N \models \chi(c) \implies M \models \chi(c) $$

As $$\phi(x)$$, $$\psi(x)$$, and $$\chi(x)$$ are all equivalent, we obtain the desired (*) above.

QED.