« Section 2.7: Problem 2 Solution

# Section 2.7: Problem 3 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
Show that an interpretation of a complete theory into a satisfiable theory is faithful.
Let $T$ be a complete theory in $L$ , $T'$ be a satisfiable theory in $L'$ , and $\pi$ be an interpretation of $T$ into $T'$ . Then, by the very first paragraph on page 169, $\pi^{-1}[T']$ is satisfiable, by the definition of an interpretation of a theory into another theory, $T\subseteq\pi^{-1}[T']$ , and by Exercise 2 of Section 2.6, $T=\pi^{-1}[T']$ , i.e. $\pi$ is faithful.