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Math Help - Model Theory - definiable classes

  1. #1
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    Model Theory - definiable classes

    Hi,

    This time I've problem with following exercise:

    Let L be a first-order language and for each i \in I let \mathbb{K}_i be a class of L-structures.
    Show that \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 A is a model of \Phi, or that \Phi is true in A, when A \models \Phi holds. Given a theory T in L_{\infty \omega}, we say that A is a model of T, in symbols A \models T, if A is a model of every sentence in T.
    Let L be a language and \mathbb{K} a class of L-structures. We define the L-theory of \mathbb{K}, Th_L(\mathbb{K}), to be the set (or class) of all sentences \Phi of L such that A \models \Phi for every structure A in \mathbb{K}. We omit the subscript _L when L is first-order: the theory of \mathbb{K}, \mbox{Th}(\mathbb{K}), is the set of all first-order sentences which are true in every structure in \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 \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 \{\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 \mbox{Th} (\bigcup_{i \in I}\mathbb{K}_i)\subseteq\mbox{Th}(\mathbb{K}_i) for every i, and that \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:

    \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 \Phi\in\mbox{Th} (\bigcup_{j \in I}\mathbb{K}_j). By definition of Th, \Phi is true in every structure of \bigcup_{j \in I}\mathbb{K}_j. In particular, \Phi is true in every structure of \mathbb{K}_i, i.e., \Phi\in\mbox{Th}(\mathbb{K}_i).
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