Universal theory

A universal formula is a formula $$\phi(x)$$ of the form
 * $$\forall y : \psi(x;y)$$

with $$\psi(x;y)$$ quantifier-free. (Here, x and y are tuples).

A universal theory is a theory consisting of universal sentences. Give a structure M, the universal theory of M denotes the set of all universal sentences true in M. More generally, if C is a class of structures in some language, then the universal theory of C  is the set of all universal sentences true in all structures in C.

If T is a theory, T∀ denotes the set of all universal sentences implied by T. If T is the elementary theory of a class of structures C, then T∀ is the universal theory of C.

Similarly, an existential formula is a formula $$\phi(x)$$ of the form
 * $$\exists y : \psi(x;y)$$

with $$\psi(x;y)$$ quantifier-free. One could also define the notion of "existential theory" but this turns out to not be particularly useful.

Important Facts

 * If T is a universal theory and M is a model of T, any substructure of M is a model of T.
 * Conversely, suppose T is a theory with the property that every substructure of a model of T is also a model of T. Then T is equivalent to a universal theory.
 * If T is any theory, then a structure M is a model of T∀ if and only if M embeds as a substructure into a model of T.

On the level of formulas, worthwhile things to know are:
 * Positive boolean combinations of universal formulas are (equivalent to) universal formulas. Positive boolean combinations of existential formulas are existential formulas.
 * Negations of universal formulas are existential, and vice versa.
 * Existential formulas are preserved upwards in inclusions of structures: if M ⊆ N is an inclusion of structures, a is a tuple from M, and $$\phi(x)$$ is an existential formula, then
 * $$ M \models \phi(a) \implies N \models \phi(a) $$


 * Universal formulas are preserved downwards in inclusions of structures: if M ⊆ N is an inclusion of structures, a is a tuple from M, and $$\phi(x)$$ is a universal formula, then
 * $$ N \models \phi(a) \implies M \models \phi(a) $$


 * Conversely, suppose that $$\phi(x)$$ is preserved upwards or downwards in inclusions of structures. Then $$\phi(x)$$ is equivalent to an existential or universal formula, respectively.
 * More generally, suppose that T is a theory and $$\phi(x)$$ is preserved upwards in inclusions of models of T. In other words, suppose that whenever M ⊆ N is an inclusion of models of T, and a is a tuple from M, we have
 * $$ M \models \phi(a) \implies N \models \phi(a)$$
 * THEN $$\phi(x)$$ is equivalent mod T to an existential formula. That is, there is an existential formula $$\phi'(x)$$ such that
 * $$T \vdash \forall x : \phi(x) \leftrightarrow \phi'(x)$$


 * Similarly, if T is a theory and $$\phi(x)$$ is preserved downwards in models of T, then $$\phi(x)$$ is equivalent mod T to a universal formula.

These last two points play a key role in the proof that the different characterizations of model completeness are equivalent.

Proofs
The negation $$ \neg \forall y : \psi(x;y)$$ of a universal formula is equivalent by De Morgan's laws to the existential formula $$\exists y : \neg \psi(x;y)$$. Similarly, the negation of an existential formula is equivalent to a universal formula.

Existential formulas are closed under positive boolean combinations because of the equivalences
 * $$ \left(\exists y: \phi(x;y)\right) \wedge \left(\exists z : \psi(x;z)\right)

\iff \exists y \exists z : \phi(x;y) \wedge \psi(x;z) $$
 * $$ \left(\exists y: \phi(x;y)\right) \vee \left(\exists z : \psi(x;z)\right)

\iff \exists y \exists z : \phi(x;y) \vee \psi(x;z) $$ By De Morgan's laws, the same holds for universal formulas.

Lemma 1: Existential statements are preserved upwards in inclusions: if M ⊆ N, a is a tuple from M, $$\phi(x)$$ is existential, and $$M \models \phi(a)$$, then $$N \models \phi(a)$$. Similarly, universal statements are preserved downwards.

Proof: Write $$\phi(x)$$ as $$\exists y : \psi(x;y)$$. Since $$M \models \phi(a)$$, we have $$M \models \psi(a;b)$$ for some tuple b from M. As $$\psi(x;y)$$ is quantifier-free, $$\psi(a;b)$$ has the same truth value in M as in N. (The ambient model only affects the interpretation of quantifiers.) Therefore, $$N \models \psi(a;b)$$ holds, so
 * $$ N \models \exists y : \psi(a;y)$$,

i.e., $$N \models \phi(a)$$. So existential formulas are preserved upwards in inclusions.

Meanwhile, if $$\chi(x)$$ is a universal formula, then $$\neg \chi(x)$$ is an existential formula. Since $$\neg \chi$$ is preserved upwards, $$\chi$$ is preserved downwards (this is the contrapositive). QED.

In particular, if M ⊆ N and M satisfies some existential sentence, then so does N. And if N satisfies some universal sentence, then so does M. Therefore we have:

Corollary 2: If T is a universal theory, then any substructure of a model of T is a model of T.

Lemma 3: Let T be a theory, and suppose M is a model of T∀. Then M embeds into a model of T.

Proof: Let L be the language of T. Let $$T'$$ be the L(M)-theory consisting of the union of T with the diagram of M. If $$T'$$ is consistent, it has a model N. Then N is a model of the diagram of M, so M is a substructure of N. Also, N is a model of T. So we are done.

Suppose therefore that $$T'$$ is not consistent. By compactness, some finite subset of $$T'$$ is inconsistent. The part of this finite subset coming from the diagram of M can be expressed by a single statement of the form $$\chi(a)$$, where a is a tuple from M, $$\chi(x)$$ is a quantifier-free L-formula, and $$M \models \chi(a)$$.

Then we are saying that $$T \cup \{\chi(a)\}$$ is inconsistent. By the Lemma on Constants,
 * $$ T \vdash \forall x : \neg \chi(x)$$

In particular, $$\forall x : \neg \chi(x)$$ is part of T∀. Therefore it must hold in M, by assumption. But
 * $$ M \models \forall x : \neg \chi(x)$$ implies $$ M \models \neg \chi(a)$$

contradicting the fact that $$M \models \chi(a)$$. QED.

Corollary 4: Suppose T is a theory with the property that every substructure of a model of T is itself a model of T. Then T is equivalent to a universal theory. In fact, T is equivalent to T∀.

Proof: Indeed, any model of T embeds into a model of T (namely, itself), and so is a model of T∀. Conversely, if M is a model of T∀, then M embeds into a model N of T. But then M is a substructure of a model of T, so by assumption, M is itself a model of T. Therefore, models of T are the same thing as models of T∀. QED.

We have proven all the claims above except

Theorem 5: Let T be a theory (possibly the empty theory). Let $$\phi(x)$$ be a formula. Suppose that $$\phi(x)$$ is preserved upwards in models of T, i.e., if M ⊆ N is an inclusion of models of T, and a is a tuple from M, then
 * $$ M \models \phi(a) \implies N \models \phi(a)$$

Then there is an existential formula $$\phi'(x)$$ which is equivalent to $$\phi(x)$$ mod T. Similarly, any formula which is preserved downwards in inclusions of models of T is equivalent to a universal formula.

Proof:

First we show that whenever $$\phi(x)$$ holds, it holds because of an existential formula.

Claim: If M is a model of T and $$ M \models \phi(a)$$ for some a, then there is an existential formula $$\psi(x)$$ such that
 * $$ T \vdash \forall x : \psi(x) \rightarrow \phi(x)$$, i.e., $$\psi$$ implies $$\phi$$ mod T

and
 * $$ M \models \psi(a)$$.

Proof: Let $$T'$$ consist of the the union of T, the diagram of M, and the statement $$\neg \phi(a)$$. If $$T'$$ has a model N, then N is a model of T extending M, in which $$\neg \phi(a)$$ holds. This contradicts the hypothesis that $$\phi(x)$$ is preserved upwards in inclusions of models of T.

Therefore $$T'$$ is inconsistent. By compactness, this inconsistency must be witnessed by a finite part of the diagram of M. Therefore, there is some quantifier-free formula $$\chi(x,y)$$ and some b in M such that $$M \models \chi(a,b)$$ and
 * $$ T \cup \{\neg \phi(a)\} \cup \{\chi(a,b)\}$$

is inconsistent. By the Lemma on constants,
 * $$ T \vdash \forall x, y : \chi(x,y) \rightarrow \phi(x)$$

Equivalently,
 * $$ T \vdash \forall x : (\exists y : \chi(x;y)) \rightarrow \phi(x)$$

Now let $$\psi(x)$$ be the formula $$\exists y : \chi(x;y)$$. This is an existential formula, and it implies $$\phi(x)$$, mod T. Also, $$M \models \psi(a)$$, by taking y = c. So we have proven the claim. QEDclaim.

Now let $$\Psi(x)$$ be the set of all formulas $$\neg \psi(x)$$, where $$\psi(x)$$ is existential and implies $$\phi(x)$$.

Consider the theory
 * $$T'' = T \cup \{\phi(a)\} \cup \Psi(a)$$

where a is a new constant. Suppose $$T$$ has a model M, and let a be the interpretation of the symbol a in M''. Then M is a model of T, and
 * $$ M \models \phi(a)$$.

By the claim, $$\phi(a)$$ must hold because of some existential formula: there must be an existential formula $$\psi(x)$$ which implies $$\phi(x)$$, with $$\psi(a)$$ holding in M. But then $$\neg \psi(x)$$ is one of the formulas in $$\Psi(x)$$. Since $$M \models T \supset \Psi(a)$$, we have $$M \models \neg \psi(a)$$, contradicting the fact that $$\psi(a)$$ holds in M.

In other words, the Claim means exactly that $$T''$$ is inconsistent. Now by compactness, some finite subset of $$T''$$ is inconsistent. Therefore we can find finitely many existential formulas $$\psi_1(x),\ldots,\psi_n(x)$$, each implying $$\phi(x)$$, such that
 * $$ T \cup \{\phi(a), \neg \psi_1(a), \ldots, \neg \psi_n(a)\}$$

is inconsistent. By the Lemma on Constants, this means that
 * $$ T \vdash \forall x : \phi(x) \rightarrow \bigvee_{i = 1}^n \psi_i(x)$$

So, mod T, $$\phi(x)$$ implies the formula $$\bigvee_{i = 1}^n \psi_i(x)$$. Conversely, since each $$\psi_i(x)$$ implies $$\phi(x)$$, so does their disjunction $$\bigvee_{i = 1}^n \psi_i(x)$$. Therefore,
 * $$\phi(x)$$ is equivalent to $$\bigvee_{i = 1}^n \psi_i(x)$$, mod T.

But $$\bigvee_{i = 1}^n \psi_i(x)$$ is a disjunction of existential formulas, so it is itself an existential formula. Thus $$\phi(x)$$ is equivalent to an existential formula. This completes the proof of the first claim of the Theorem.

For the other, suppose that $$\phi(x)$$ is a formula which is preserved downwards in inclusions of models of T. Then, contrapositively, $$\neg \phi(x)$$ is preserved upwards. So, by what we have shown, $$\neg \phi(x)$$ is equivalent to an existential formula. Then $$\phi(x)$$ is equivalent to the negation of an existential formula, i.e., to a universal formula. QED.