Tarski-Vaught Test
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Theorem
Let $\MM, \NN$ be $\LL$-structures such that $\MM$ is a substructure of $\NN$.
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Then $\MM$ is an elementary substructure of $\NN$ if and only if:
- 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 $\NN \models \map \phi {m, \bar a}$.
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The condition on the right side of the if and only if 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 $\NN$ if and only if:
- 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 if and only if 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 $\MM$.
Proof
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.
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$\Box$
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:
- $\MM \models \map {\psi_0} {\bar a} \wedge \map {\psi_1} {\bar a}$
- $\MM \models \map {\psi_0} {\bar a}$
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 $\NN \models \map {\psi_1} {\bar a}$
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.
$\blacksquare$
Proof of Generalization
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.
$\Box$
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.
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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 $\MM \models \map {\psi_1} {\bar a}$
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 $\NN \models \map {\psi_1} {\bar a}$
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.
$\blacksquare$
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Generalization
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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.
$\blacksquare$
Source of Name
This entry was named for Alfred Tarski and Robert Lawson Vaught.