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Thread: Model Theory - definiable classes

  1. #1
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    Model Theory - definiable classes

    Hi,

    This time I've problem with following exercise:

    Let $\displaystyle L$ be a first-order language and for each $\displaystyle i \in I$ let $\displaystyle \mathbb{K}_i$ be a class of $\displaystyle L$-structures.
    Show that $\displaystyle \mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i) =\bigcap_{i \in I} \mbox{Th}(\mathbb{K}_i)$.

    Here are some basic definitions from the book "Model Theory", W. Hodges (which could help):

    A sentence is a formula with no free variables. A theory is a set of sentences.
    We say that $\displaystyle A$ is a model of $\displaystyle \Phi$, or that $\displaystyle \Phi$ is true in $\displaystyle A$, when $\displaystyle A \models \Phi$ holds. Given a theory $\displaystyle T$ in $\displaystyle L_{\infty \omega}$, we say that $\displaystyle A$ is a model of $\displaystyle T$, in symbols $\displaystyle A \models T$, if $\displaystyle A$ is a model of every sentence in $\displaystyle T$.
    Let $\displaystyle L$ be a language and $\displaystyle \mathbb{K}$ a class of $\displaystyle L$-structures. We define the $\displaystyle L$-theory of $\displaystyle \mathbb{K}$, $\displaystyle Th_L(\mathbb{K})$, to be the set (or class) of all sentences $\displaystyle \Phi$ of $\displaystyle L$ such that $\displaystyle A \models \Phi$ for every structure $\displaystyle A$ in $\displaystyle \mathbb{K}$. We omit the subscript $\displaystyle _L$ when $\displaystyle L$ is first-order: the theory of $\displaystyle \mathbb{K}$, $\displaystyle \mbox{Th}(\mathbb{K})$, is the set of all first-order sentences which are true in every structure in $\displaystyle \mathbb{K}$.

    Thanks for any help.
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  2. #2
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    Quote Originally Posted by Arczi1984 View Post
    Show that $\displaystyle \mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i) =\bigcap_{i \in I} \mbox{Th}(\mathbb{K}_i)$.
    Consider two structures A and B. Then $\displaystyle \{\Phi\mid A\models\Phi\text{ and }B\models\Phi\}=\{\Phi\mid A\models\Phi\}\cap\{\Phi\mid B\models\Phi\}$. Thus, both sides in the quote above consist of sentences that are true in all structures.

    In more detail, show that $\displaystyle \mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i)\subseteq\mbox{Th}(\mathbb{K}_i)$ for every $\displaystyle i$, and that $\displaystyle \bigcap_{i \in I} \mbox{Th}(\mathbb{K}_i)\subseteq\mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i)$.
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  3. #3
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    I understand the first part. But how can I show, for example:

    $\displaystyle \mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i)\subseteq\mbox{Th}(\mathbb{K}_i)$

    using this fact?
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  4. #4
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    Suppose $\displaystyle \Phi\in\mbox{Th} (\bigcup_{j \in I}\mathbb{K}_j)$. By definition of Th, $\displaystyle \Phi$ is true in every structure of $\displaystyle \bigcup_{j \in I}\mathbb{K}_j$. In particular, $\displaystyle \Phi$ is true in every structure of $\displaystyle \mathbb{K}_i$, i.e., $\displaystyle \Phi\in\mbox{Th}(\mathbb{K}_i)$.
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