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 and define its elementary type to be the class of structures elementarily equivalent to . Show that this class is . Suggestion: Show it is .
(b) Call a class of structures elementarily closed or if whenever a structure belongs to then all elementarily equivalent structures also belong. Show that any such class is a union of classes. (A class that is a union of classes is said to be an class; this notation is derived from topology.)
(c) Conversely, show that any class that is the union of classes is elementarily closed.
(a) The suggestion uses a symbol , 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 . Now, if is in , then every sentence true in is true in (by definition of ), and for every sentence false in , is true in and, hence, in , i.e. is false in . Hence, . Vice versa, if then for every sentence , is true in iff it is true in , in particular, every sentence true in (that is every sentence in ) is true in , and is in . Therefore, an elementary type is of a set of sentences, and, hence, .
(b) Using (a), if is in , its elementary type is a subclass of , therefore, is the union of elementary types of its members, each of which is .
(c) Suppose is , and is in . Let be an class that belongs to. Then is for some set of sentences . Since belongs to , , i.e. . Hence, for every that is in , , implying that (elementary type of , according to (a)) is a subclass of . Hence, according to the definition provided in (b), , and, hence, , is elementarily closed.