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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, mot T, to a universal formula.

See below for a proof that these notions are equivalent.

ExamplesEdit

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 xy 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 equivalentEdit

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 MN 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 MN 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 MN 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

$ N \models \psi(c) \implies M \models \psi(c) $
$ M \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.