# Section 2.2: Problem 26 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
(a) Consider a fixed structure $\mathfrak{A}$ and define its elementary type to be the class of structures elementarily equivalent to $\mathfrak{A}$ . Show that this class is $EC{}_{\Delta}$ . Suggestion: Show it is $\mbox{Mod Th}\mathfrak{A}$ .
(b) Call a class $\mathcal{K}$ of structures elementarily closed or $ECL$ if whenever a structure belongs to $\mathcal{K}$ then all elementarily equivalent structures also belong. Show that any such class is a union of $EC{}_{\Delta}$ classes. (A class that is a union of $EC_{\Delta}$ classes is said to be an $EC_{\Delta\Sigma}$ class; this notation is derived from topology.)
(c) Conversely, show that any class that is the union of $EC_{\Delta}$ classes is elementarily closed.
(a) The suggestion uses a symbol $\mbox{Th}$ , which according to the List of Symbols at the end of the book is first introduced on page 152, though it is, in fact, defined on page 148. Anyway, it is clear what it means: the set of all sentences true in $\mathfrak{A}$ . Now, if $\mathfrak{B}$ is in $\mbox{Mod Th}\mathfrak{A}$ , then every sentence true in $\mathfrak{A}$ is true in $\mathfrak{B}$ (by definition of $\mbox{Mod}$ ), and for every sentence $\sigma$ false in $\mathfrak{A}$ , $\neg\sigma$ is true in $\mathfrak{A}$ and, hence, in $\mathfrak{B}$ , i.e. $\sigma$ is false in $\mathfrak{B}$ . Hence, $\mathfrak{A}\equiv\mathfrak{B}$ . Vice versa, if $\mathfrak{A}\equiv\mathfrak{B}$ then for every sentence $\sigma$ , $\sigma$ is true in $\mathfrak{A}$ iff it is true in $\mathfrak{B}$ , in particular, every sentence true in $\mathfrak{A}$ (that is every sentence in $\mbox{Th}\mathfrak{A}$ ) is true in $\mathfrak{B}$ , and $\mathfrak{B}$ is in $\mbox{Mod Th}\mathfrak{A}$ . Therefore, an elementary type is $\mbox{Mod}$ of a set of sentences, and, hence, $EC_{\Delta}$ .
(b) Using (a), if $\mathfrak{A}$ is in $\mathcal{K}$ , its elementary type is a subclass of $\mathcal{K}$ , therefore, $\mathcal{K}$ is the union of elementary types of its members, each of which is $EC_{\Delta}$ .
(c) Suppose $\mathcal{K}$ is $EC_{\Delta\Sigma}$ , and $\mathfrak{A}$ is in $\mathcal{K}$ . Let $\mathcal{L}$ be an $EC_{\Delta}$ class that $\mathfrak{A}$ belongs to. Then $\mathcal{L}$ is $\mbox{Mod}\Sigma$ for some set of sentences $\Sigma$ . Since $\mathfrak{A}$ belongs to $\mathcal{L}$ , $\vDash_{\mathfrak{A}}\Sigma$ , i.e. $\Sigma\subseteq\mbox{Th}\mathfrak{A}$ . Hence, for every $\mathfrak{B}$ that is in $\mbox{Mod Th}\mathfrak{A}$ , $\vDash_{\mathfrak{B}}\Sigma$ , implying that $\mbox{Mod Th}\mathfrak{A}$ (elementary type of $\mathfrak{A}$ , according to (a)) is a subclass of $\mbox{Mod}\Sigma=\mathcal{L}$ . Hence, according to the definition provided in (b), $\mathcal{L}$ , and, hence, $\mathcal{K}$ , is elementarily closed.