« Section 2.6: Problem 11-A Solution

# 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 $\Sigma$ is a set of sentences in a finite language (with equality), and that all models of $\Sigma$ are finite. Show that $\Sigma$ 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 $\mathbb{N}$ . Now, there is some $N$ such that all models of $\Sigma$ are of size $\le N$ . Indeed, consider $\tau_{n}=\exists v_{1}\ldots\exists v_{n}(\wedge_{1\le i , which ensures that there are at least $n$ elements in the universe. If we assume that there is a model for every finite subset of $\overline{\Sigma}=\Sigma\cup\{\tau_{n}\}_{n\in\mathbb{N}}$ , then there is a model of $\overline{\Sigma}$ , and, hence, $\Sigma$ , that must be infinite, contradicting the assumption. Therefore, for some $N$ there is no model of $\Sigma$ that would satisfy $\tau_{N+1}$ , and all models of $\Sigma$ must be of size $\le N$ . Given this, there are only finite number of models of $\Sigma$ (in the finite language) with the universe being a subset of $\mathbb{N}$ , and, according to observation 1 on page 148, every model of $\Sigma$ is isomorphic to one of them.