# Section 1.7: Problem 12 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
For each of the following conditions, give an example of an unsatisfiable set $\Gamma$ of formulas that meets the condition.
(a) Each member of $\Gamma$ is — by itself — satisfiable.
(b) For any two members $\gamma_{1}$ and $\gamma_{2}$ of $\Gamma$ , the set $\{\gamma_{1},\gamma_{2}\}$ is satisfiable.
(c) For any three members $\gamma_{1}$ , $\gamma_{2}$ , and $\gamma_{3}$ of $\Gamma$ , the set $\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ is satisfiable.
(a) $\Gamma_{1}=\{A,\neg A\}$ .
(b) $\Gamma_{2}=\{A,B,A|B\}$ .
(c) $\Gamma_{3}=\{A,B,C,\neg(A\wedge B\wedge C)\}$ .
What is the point of this exercise? Suppose we were to continue the sequence of such examples, and for every $n$ we would find an unsatisfiable set $\Gamma_{n}$ of wffs such that every its subset of size $n$ is satisfiable. Then, every subset of $\Gamma_{n}$ of size $m is also satisfiable, and $\Gamma_{n}$ can be used as an example of a set $\Gamma_{m}$ for every $m . For instance, in (a), (b) and (c), instead of providing three different sets of wffs, we could have just used the same set $\Gamma_{3}$ . Now, the question is, then, whether we can provide just one infinite set $\Gamma$ of wffs that would work as an example of a set $\Gamma_{n}$ for any $n$ . The compactness theorem says, No!