Exercise 2.5.2 Suppose that \(T\) has arbitrarily large finite models. Show that \(T\) has an infinite model.

Proof: Let \(\phi_n\) be the sentence asserting that “there exists at least \(n\) elements” which is expressible in first-order logic. Consider the theory \(T' = T \cup \{\phi_n : n \in \mathbb{N}\}\). Because \(T\) has arbitrarily large models, the theory \(T \cup \{\phi_n : n \leq N\}\) is satisfiable for any \(N\). Therefore, by compactness, \(T'\) is also satisfiable. Clearly, any model of \(T'\) is an infinite model of \(T\).