# Section 2.6: Problem 8 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
Assume that $\sigma$ is true in all infinite models of a theory $T$ . Show that there is a finite number $k$ such that $\sigma$ is true in all models $\mathfrak{A}$ of $T$ for which $|\mathfrak{A}|$ has $k$ or more elements.
If not, then for every $n$ there is some $m>n$ and a model $\mathfrak{A}_{m}$ of $T$ of size $m$ such that $\sigma$ is false in $\mathfrak{A}$ . But then, $\neg\sigma$ is true in all such $\mathfrak{A}_{m}$ ’s, and, by Theorem 26A, $T;\neg\sigma$ has an infinite model $\mathfrak{A}$ , implying that $\sigma$ is false in the infinite model $\mathfrak{A}$ of $T$ . Contradiction.