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If first-order formula is true in every expansion of some structure, is it true in any expansion of any elementarily equivalent structure?

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2

$begingroup$

Let $M$ be a $Sigma$-structure and $pnotinSigma$ be a predicate symbol. Let $T := Th(M)$ in $Sigma$. Let $phi$ be a formula in $Sigma(p)$ with the following property: for all $Sigma(p)$-structures N,

beginalign
textif NmathordupharpoonrightSigma = Mtext, then NvDashphi.
endalign

In other words, let every $Sigma(p)$-expansion of $M$ be a model of $phi$. If we consider $T$ as a set of $Sigma(p)$-sentences, I wonder if

beginalign
TvDashphi.
endalign

In other words, if $p$ "occurs tautologically" in $phi$ with respect to $M$, does it hold for every elementarily equivalent to $M$ $Sigma$-structure?

first-order-logic model-theory

share|cite|improve this question

asked Mar 22 at 0:02

dvvrddvvrd

1225

$endgroup$

add a comment | 

2

$begingroup$

Let $M$ be a $Sigma$-structure and $pnotinSigma$ be a predicate symbol. Let $T := Th(M)$ in $Sigma$. Let $phi$ be a formula in $Sigma(p)$ with the following property: for all $Sigma(p)$-structures N,

beginalign
textif NmathordupharpoonrightSigma = Mtext, then NvDashphi.
endalign

In other words, let every $Sigma(p)$-expansion of $M$ be a model of $phi$. If we consider $T$ as a set of $Sigma(p)$-sentences, I wonder if

beginalign
TvDashphi.
endalign

In other words, if $p$ "occurs tautologically" in $phi$ with respect to $M$, does it hold for every elementarily equivalent to $M$ $Sigma$-structure?

first-order-logic model-theory

share|cite|improve this question

asked Mar 22 at 0:02

dvvrddvvrd

1225

$endgroup$

add a comment | 

$begingroup$

Let $M$ be a $Sigma$-structure and $pnotinSigma$ be a predicate symbol. Let $T := Th(M)$ in $Sigma$. Let $phi$ be a formula in $Sigma(p)$ with the following property: for all $Sigma(p)$-structures N,

beginalign
textif NmathordupharpoonrightSigma = Mtext, then NvDashphi.
endalign

In other words, let every $Sigma(p)$-expansion of $M$ be a model of $phi$. If we consider $T$ as a set of $Sigma(p)$-sentences, I wonder if

beginalign
TvDashphi.
endalign

In other words, if $p$ "occurs tautologically" in $phi$ with respect to $M$, does it hold for every elementarily equivalent to $M$ $Sigma$-structure?

first-order-logic model-theory

share|cite|improve this question

asked Mar 22 at 0:02

dvvrddvvrd

1225

$endgroup$

share|cite|improve this question

asked Mar 22 at 0:02

dvvrddvvrd

1225

share|cite|improve this question

asked Mar 22 at 0:02

dvvrddvvrd

1225

asked Mar 22 at 0:02

dvvrddvvrd

1225

asked Mar 22 at 0:02

dvvrddvvrd

1225

1 Answer
1

active

oldest

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2

$begingroup$

Here is a counterexample. Let $Sigma = leq$, and consider the $Sigma(p)$-sentence $varphi$ asserting that if $p$ defines a non-empty set with an upper bound, then the set defined by $p$ has a least upper bound. Then $varphi$ is true in every $Sigma(p)$-expansion of $(mathbbR,leq)$, since this is a complete linear order, but it is not true in every $Sigma(p)$-expansion of every structure elementarily equivalent to $(mathbbR,leq)$. For example, if we interpret $p$ as $(-infty,sqrt2)$ in $(mathbbQ,leq)$, $varphi$ is false.

On the other hand, if $varphi$ satisfies your desired conclusion, i.e. if $varphi$ is true in every $Sigma(p)$-expansion of every model of $T = textTh(M)$, then it follows immediately from Beth's definability theorem that $varphi$ is equivalent modulo $T$ to a $Sigma$-sentence.

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answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758

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2

$begingroup$

Here is a counterexample. Let $Sigma = leq$, and consider the $Sigma(p)$-sentence $varphi$ asserting that if $p$ defines a non-empty set with an upper bound, then the set defined by $p$ has a least upper bound. Then $varphi$ is true in every $Sigma(p)$-expansion of $(mathbbR,leq)$, since this is a complete linear order, but it is not true in every $Sigma(p)$-expansion of every structure elementarily equivalent to $(mathbbR,leq)$. For example, if we interpret $p$ as $(-infty,sqrt2)$ in $(mathbbQ,leq)$, $varphi$ is false.

On the other hand, if $varphi$ satisfies your desired conclusion, i.e. if $varphi$ is true in every $Sigma(p)$-expansion of every model of $T = textTh(M)$, then it follows immediately from Beth's definability theorem that $varphi$ is equivalent modulo $T$ to a $Sigma$-sentence.

share|cite|improve this answer

answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758

$endgroup$

add a comment | 

2

$begingroup$

Here is a counterexample. Let $Sigma = leq$, and consider the $Sigma(p)$-sentence $varphi$ asserting that if $p$ defines a non-empty set with an upper bound, then the set defined by $p$ has a least upper bound. Then $varphi$ is true in every $Sigma(p)$-expansion of $(mathbbR,leq)$, since this is a complete linear order, but it is not true in every $Sigma(p)$-expansion of every structure elementarily equivalent to $(mathbbR,leq)$. For example, if we interpret $p$ as $(-infty,sqrt2)$ in $(mathbbQ,leq)$, $varphi$ is false.

On the other hand, if $varphi$ satisfies your desired conclusion, i.e. if $varphi$ is true in every $Sigma(p)$-expansion of every model of $T = textTh(M)$, then it follows immediately from Beth's definability theorem that $varphi$ is equivalent modulo $T$ to a $Sigma$-sentence.

share|cite|improve this answer

answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758

$endgroup$

add a comment | 

$begingroup$

Here is a counterexample. Let $Sigma = leq$, and consider the $Sigma(p)$-sentence $varphi$ asserting that if $p$ defines a non-empty set with an upper bound, then the set defined by $p$ has a least upper bound. Then $varphi$ is true in every $Sigma(p)$-expansion of $(mathbbR,leq)$, since this is a complete linear order, but it is not true in every $Sigma(p)$-expansion of every structure elementarily equivalent to $(mathbbR,leq)$. For example, if we interpret $p$ as $(-infty,sqrt2)$ in $(mathbbQ,leq)$, $varphi$ is false.

On the other hand, if $varphi$ satisfies your desired conclusion, i.e. if $varphi$ is true in every $Sigma(p)$-expansion of every model of $T = textTh(M)$, then it follows immediately from Beth's definability theorem that $varphi$ is equivalent modulo $T$ to a $Sigma$-sentence.

share|cite|improve this answer

answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758

$endgroup$

share|cite|improve this answer

answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758

answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758

answered Mar 22 at 6:45

Alex KruckmanAlex Kruckman

28.5k32758