# 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.