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urpzilmöräqÜ
urpzilmöräqÜ
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Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite structures$\mathcal L$-structures such that for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\iff\mathcal B\models\varphi.$$

Does it follow that $\mathcal A$ and $\mathcal B$ are isomorphic?

Clearly, for finite signatures $\mathcal L$ the answer would be "yes".

Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite structures such that for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\iff\mathcal B\models\varphi.$$

Does it follow that $\mathcal A$ and $\mathcal B$ are isomorphic?

Clearly, for finite signatures $\mathcal L$ the answer would be "yes".

Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite $\mathcal L$-structures such that for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\iff\mathcal B\models\varphi.$$

Does it follow that $\mathcal A$ and $\mathcal B$ are isomorphic?

Clearly, for finite signatures $\mathcal L$ the answer would be "yes".

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urpzilmöräqÜ
urpzilmöräqÜ
  • 123
  • 3

$\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ for finite $\mathcal L$-structures where $\mathcal L$ is an infinite signature

Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite structures such that for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\iff\mathcal B\models\varphi.$$

Does it follow that $\mathcal A$ and $\mathcal B$ are isomorphic?

Clearly, for finite signatures $\mathcal L$ the answer would be "yes".