Model Theory - definiable classes

**Arczi1984** · November 17th 2010, 04:12 AM

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.

**emakarov** · November 17th 2010, 04:40 AM

Originally Posted by Arczi1984

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

**Arczi1984** · November 19th 2010, 08:22 AM

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?

**emakarov** · November 19th 2010, 09:21 AM

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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