A theory *T* is **inductive** if any union of an increasing chain of models of *T* (possibly of transfinite length) is a model of *T* as well.

An **∀∃-sentence** is a sentence of the form

with *x* and *y* tuples and quantifier-free. An **∀∃-theory** is a theory made of ∀∃-sentences.

It turns out that these two notions are essentially the same:

**Theorem:** Let *T* be a theory. The following are equivalent:

(a) If is an increasing chain of models of *T* (so for ), then the union is a model of *T*.

(b) If is an increasing chain of models of *T*, of length , then is a model of *T*.

(c) *T* is equivalent to an ∀∃-theory.

*Proof:* Clearly (a) implies (b), since (b) is the case of (a). The implication from (c) to (a) is also relatively straightforward. It boils down to the fact that if is an increasing chain of structures, and

for all , then

Indeed, if *x = a* is a tuple from the union, then *a* is finite, so *a* comes from some specific . By (*), there is some *b* in such that . As is quantifier-free, also holds in the union. So for any *a* from the union, there is some *b* from the union such that holds, as claimed.

It remains to show that (b) implies (c). Let *T*_{∀∃} denote the set of all ∀∃-sentences implied by *T*. Any model of *T* is a model of *T*_{∀∃}. If we show the converse, then *T* is equivalent to the ∀∃-theory *T*_{∀∃}, and we are done.

**Claim:** Let *M* be a model of *T*_{∀∃}. Then there is a model *M*_{2} of *T* extending *M*, and a structure *M*_{3} extending *M*_{2} such that *M*_{3} is an elementary extension of *M*.

*Proof:* Elementary extensions of *M* are the same thing as models of the elementary diagram of *M*. Let *S* be the elementary diagram of *M*. Consider *S*_{∀}, the set of universal sentences over *M* which hold in *M*. If *S*_{∀} ∪ *T* is inconsistent, then there is some true universal statement about some element of *M* which contradicts *T*. That is, there is a universal formula , and a tuple *a* from *M*, such that

and

is inconsistent.

By the lemma on constants,

Now the formula is existential, so the sentence is an ∀∃-sentence. In particular, it is part of *T*_{∀∃}, so it holds in *M*:

But this contradicts the fact that .

Therefore *S*_{∀} ∪ *T* actually is consistent. Let *M*_{2} be a model. Then *M*_{2} is a model of *T*. Also, *S*_{∀} contains all the quantifier-free statements in *S*, which exactly constitute the diagram of *M*. So *M*_{2} is a model of the diagram of *M*, i.e., *M*_{2} is an extension of *M*.

Finally, since *M*_{2} satisfies the universal theory of the elementary diagram of *M*, we can embed *M*_{2} into some model *M*_{3} of the elementary diagram of *M*. But a model of the the elementary diagram of *M* is just an elementary extension of *M*. So we have produced the desired extensions. QED_{claim}

Now, given a model *M* of *T*_{∀∃}, we will prove that *M* is a model of *T*, completing the proof of the Theorem.

By the Claim, we can produce

with and . Since , is itself a model of . So we can apply the claim to , producing

with and .

Continuing on in this fashion, one gets an infinite ascending chain of structures

such that and

.

By the Tarski-Vaught Theorem,

.

And, since

is an increasing chain of models of *T*, of length , we have

.

So *M* is an elementary substructure of a model of *T*. Therefore, *M* is a model of *T*.

So every model of *T*_{∀∃} is a model of *T*. Consequently, *T* is equivalent to the ∀∃-theory *T*_{∀∃}. This shows that (b) implies (c), completing the proof of the theorem. QED.

Many common theories have this property:

- Rings, groups, boolean algebras, anything equational
- Fields, differential fields, difference fields, integral domains
- Any model complete theory: an ascending chain of models will always be an elementary chain of models (by model completeness), so the union will still be a model.
- Any universal theory, such as
*T*_{∀}for*any*theory*T*. - DLO
- The random graph
- PAC fields

A simple example of a theory without this property is the theory of as an ordered set. Note that

is an ascending chain of models of this theory. (Each structure in this chain is isomorphic to , so certainly a model of its complete theory.) However, the union of this increasing chain is the ring , which is a dense linear order. In particular, the statement

holds in the limit, even though it is false in . So the limit is not a model of this theory.

For a more real-world example, the theory of pseudo-finite fields is not inductive. For example, let be the field with *q* elements. For each *n* > 0, let *K*_{n} be the amalgam of as *p* ranges over all primes. It is known that *K*_{n} is pseudo-finite for each *n*. Also, *K*_{n} ⊆ *K*_{n+1} for each *n*, so we have an ascending chain of pseudo-finite fields. However,

is algebraically closed, rather than pseudo-finite.

(On the other hand, the theory of pseudo-finite fields can be made to be model complete by adding in predicates SOL_{n}(*x*_{1},...,*x _{n}*) interpreted as

)

Inductive theories cooperate well with notions such as existential closedness and model companions. The main results are:

**Theorem:** Let *T* be inductive. Then every model of *T* can be embedded into an existentially closed model of *T*.

**Theorem:** Let *T* be inductive. Then *T* has a model companion if and only if the class of existentially closed models of *T* is an elementary class. If so, the models of the model companion of *T* are exactly the existentially closed models of *T*. ::: ::: :::