The Meaning of Clauses
The discovery of a non-Euclidean geometry ended a two thousand-year-old search for a refutation of the set of statements obtained by adding the negation of the Fifth Postulate (about parallel lines) to the remaining Euclidean axioms. This is an important example of a general fact: if there exists a possible world in which a set of statements is true, then there does not exist a refutation of this set of statements. The reverse implication is a fundamental result of logic, the completeness theorem of Gödel, for the proof of which we cannot content ourselves with a generic intuition. Instead, we have to learn to work with some fundamental concepts that will be defined in this chapter.