# Section 2.6: Problem 12-A 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

*(Additional)*Assume that is a set of sentences in a finite language (with equality), and that all models of are finite. Show that has only finitely many models, up to isomorphism.

“Up to isomorphism” means that we can restrict our attention to the models with the universe being a subset of
. Now, there is some
such that all models of
are of size
. Indeed, consider
, which ensures that there are at least
elements in the universe. If we assume that there is a model for every finite subset of
, then there is a model of
, and, hence,
, that must be infinite, contradicting the assumption. Therefore, for some
there is no model of
that would satisfy
, and all models of
must be of size
. Given this, there are only finite number of models of
(in the finite language) with the universe being a subset of
, and, according to observation 1 on page 148, every model of
is isomorphic to one of them.