“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 $\tau_{n}=\exists v_{1}\ldots\exists v_{n}(\wedge_{1\le i<j\le n}v_{i}\neq v_{j})$ , 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.