David Marker, Model Theory: An Introduction, Exercise 2.5.2
mathematics
model theory
textbook solutions
A compactness proof that a theory with arbitrarily large finite models has an infinite model.
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
\[ \exists x_1 \exists x_2 \cdots \exists x_n \, \bigwedge_{i \neq j} x_i \neq x_j \]
asserting that “there exist at least \(n\) elements.” Consider the theory
\[ T' = T \cup \{\phi_n : n \in \mathbb{N}^+\}. \]
Because \(T\) has arbitrarily large finite models, every finite subset of \(T'\) is satisfiable. Therefore, by compactness, \(T'\) is also satisfiable. Clearly, any model of \(T'\) is an infinite model of \(T\). \(\square\)