Tarski-Vaught Test

Theorem
Let $\MM, \NN$ be $\LL$-structures such that $\MM$ is a substructure of $\NN$.

Then $\MM$ is an elementary substructure of $\NN$ :
 * for every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$:
 * if there exists an $n$ in $\nN$ such that $\NN \models \map \phi {n, \bar a}$
 * then there exists an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$.

The condition on the right side of the statement above can be rephrased as:


 * Every existential statement with parameters from $\MM$ which is satisfied in $\NN$ can be witnessed by an element from the substructure $\MM$.

Generalization
Let $\NN$ be an $\LL$-structure, and $\MM \subseteq \NN$.

Then $\MM$ is the universe of an elementary substructure of $\NN$ :
 * for every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$
 * if there exists an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$
 * then there exists an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$.

The condition on the right side of the statement above can be rephrased as::
 * Every existential statement with parameters from $\MM$ which is satisfied in $\NN$ can be witnessed by an element from the subset $\MM$.

Sufficient Condition
Let $\MM$ be an elementary substructure of $\NN$.

Then $\NN \models \exists x: \map \phi {x, \bar a}$ implies that $\MM \models \exists x: \map \phi {x, \bar a}$.

Hence there exists some $m$ in $\MM$ such that:
 * $\MM \models \map \phi {m, \bar a}$.

Passing back up to $\NN$ yields the result.

Necessary Condition
Let $\MM$ be such that:


 * For every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$:
 * if there exists an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$
 * then there exists an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$.

It is to be shown that $\MM$ is an elementary substructure of $\NN$.

That is:
 * for every $\LL$-formula $\map \psi {\bar v}$

and:
 * for every $\bar a$ in $\MM$:
 * $\paren {\MM \models \map \phi {\bar a} } \iff \paren {\NN \models \map \phi {\bar a} }$

The proof proceeds by induction on complexity of formulas.

Let $\psi$ be quantifier free.

From Quantifier Free Formula is Preserved by Superstructure, quantifier free formulas with parameters from $\MM$ are preserved when passing to and from superstructures.

Hence the result holds.

Let the result holds for $\psi$.

Consider $\neg \psi$:

We have that:
 * $\paren {\MM \models \neg \map \psi {\bar a} } \iff \paren {\MM \not \models \map \psi ({\bar a} }$

By the inductive hypothesis:
 * $\paren {\MM \not \models \map \psi {\bar a} } \iff \paren {\NN \not \models \map \psi {\bar a} }$

Thus the result follows for $\neg \psi$.

Let the result hold for $\psi_0$ and $\psi_1$.

Consider $\psi_0 \wedge \psi_1$:

We have:


 * $\MM \models \map {\psi_0} {\bar a} \wedge \map {\psi_1} {\bar a}$


 * $\MM \models \map {\psi_0} {\bar a}$
 * $\MM \models \map {\psi_0} {\bar a}$

and:
 * $\MM \models \map {\psi_1} {\bar a}$

By the inductive hypothesis:
 * $\MM \models \map {\psi_0} {\bar a}$ and $\MM \models \map {\psi_1} {\bar a}$


 * $\NN \models \map {\psi_0} {\bar a}$ and $\NN \models \map {\psi_1} {\bar a}$
 * $\NN \models \map {\psi_0} {\bar a}$ and $\NN \models \map {\psi_1} {\bar a}$

Thus the result follows for $\psi_0 \wedge \psi_1$.

Let the result holds for $\psi$.

Consider:
 * $\exists x: \map \psi x$

We prove the two directions of this case separately.

First let:
 * $\MM \models \exists x: \map \psi {x, \bar a}$

Then there exists $m \in \MM$ such that:
 * $\MM \models \map \psi {m, \bar a}$

By the inductive hypothesis:
 * $\NN \models \map \psi {m, \bar a}$

and so:
 * $\NN \models \exists x: \map \psi {x, \bar a}$

Conversely, let:
 * $\NN \models \exists x: \map \psi {x, \bar a}$

By assumption, there exists $m \in \MM$ such that:
 * $\NN \models \map \psi {m, \bar a}$

By the inductive hypothesis:
 * $\MM \models \map \psi {m, \bar a}$

and hence:
 * $\MM \models \exists x: \psi {x, \bar a}$

completing the proof.

Sufficient Condition
Let $\MM$ be an elementary substructure of $\NN$.

Then:
 * $\paren {\NN \models \exists x: \map \phi {x, \bar a} } \implies \paren {\MM \models \exists x: \map \phi {x, \bar a} }$

Hence there exists $m \in \MM$ such that:
 * $\MM \models \map \phi {m, \bar a}$

Passing back up to $\NN$ yields the result.

Necessary Condition
Let $\MM$ be such that:


 * for every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$:
 * if there exists an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$
 * then there exists an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$.

It is easily shown, using formulae of the form $\map \phi {x, \bar v} = \map f {\bar v} = x$, that $\MM$ is closed under functions, and hence the universe of a substructure.

It remains to be shown that $\MM$ is an elementary substructure of $\NN$.

That is:
 * for every $\LL$-formula $\map \psi {\bar v}$ and for every $\bar a$ in $\MM$:
 * $\paren {\MM \models \map \phi {\bar a} } \iff \paren {\NN \models \map \phi {\bar a} }$

The proof proceeds by induction on complexity of formulas.

Let $\psi$ is quantifier free.

From Quantifier Free Formula is Preserved by Superstructure, quantifier free formulas with parameters from $\MM$ are preserved when passing to and from superstructures.

Hence the result holds.

Let the result hold for $\psi$.

Consider $\neg \psi$:

We have:
 * $\paren {\MM \models \neg \map \psi {\bar a} } \iff \paren {\MM \not \models \map \psi {\bar a} }$

By the inductive hypothesis:
 * $\paren {\MM \not \models \map \psi {\bar a} } \iff \paren {\NN \not \models \map \psi {\bar a} }$

Hence the result follows for $\neg \psi$.

Suppose the result holds for $\psi_0$ and $\psi_1$.

Consider $\psi_0 \wedge \psi_1$:

We have that:
 * $\MM \models \map {\psi_0} {\bar a} \wedge \map {\psi_1} {\bar a}$


 * $\MM \models \map {\psi_0} {\bar a}$ and $\MM \models \map {\psi_1} {\bar a}$
 * $\MM \models \map {\psi_0} {\bar a}$ and $\MM \models \map {\psi_1} {\bar a}$

By the inductive hypothesis:
 * $\MM \models \map {\psi_0} {\bar a}$ and $\MM \models \map {\psi_1} {\bar a}$


 * $\NN \models \map {\psi_0} {\bar a}$ and $\NN \models \map {\psi_1} {\bar a}$
 * $\NN \models \map {\psi_0} {\bar a}$ and $\NN \models \map {\psi_1} {\bar a}$

The result then follows for $\psi_0 \wedge \psi_1$.

Let the result hold for $\psi$.

Consider $\exists x: \map \psi x$:

We prove the two directions of this case separately.

First let:
 * $\MM \models \exists x: \map \psi {x, \bar a}$

Then there exists $m \in \MM$ such that:
 * $\MM \models \map \psi {m, \bar a}$

By the inductive hypothesis:
 * $\NN \models \map \psi {m, \bar a}$

and so:
 * $\NN \models \exists x: \map \psi {x, \bar a}$

Conversely, let:
 * $\NN \models \exists x: \map \psi {x, \bar a}$

Then by assumption, there exists $m \in \MM$ such that:
 * $\NN \models \map \psi {m, \bar a}$

By the inductive hypothesis:
 * $\MM \models \map \psi {m, \bar a}$

and hence:
 * $\MM \models \exists x: \map \psi {x, \bar a}$

completing the proof.

Generalization
Another common statement of the theorem does not require $\MM$ to be a substructure of $\NN$, it is enough for it to be a subset of $\NN$.

The ($\implies$) is shown just the same as above, while the other direction easily follows, since $\MM$ satisfying the condition that for every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$, if there is an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$, then there is an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$, is closed under functions (by directly applying the condition to formulae of the form $\map \phi {x, \bar y} = \paren {x = \map f {\bar y} }$), and hence the universe of a substructure, which reduces it to the statement above.