Every statement S has a negation ¬S, read as "not S". If S is well-formed, so is its negation. For example, if we take the continuum hypothesis, i.e.: "There i

Every statement S has a negation ¬S, read as "not S". If S is well-formed, so is its negation. For example, if we take the continuum hypothesis, i.e.: "There is no set whose cardinality is strictly between that of the integers and the real numbers." and construct S to express this statement in the language of our theory (say ZFC), then the negated statement ¬S would correspond to: "There is a set whose cardinality is strictly between that of the integers and the real numbers." Famously, the continuum hypothesis is independent of ZFC. This means that (assuming consistency of ZFC), ZFC does not prove S, and also does not prove ¬S. From a consistent and syntactically complete theory, we expect that it contains exactly one of these statements. Is it the case or not? The fact that ZFC does not "tell" us which it is means that it is incomplete in this sense. Clearly, a question that is important to us is left unanswered by this theory, since neither the statement nor its negation is provable with the given axioms. A syntactically complete theory is also called deductively complete, or negation complete. An example of a first-order theory that is consistent, complete, and also decidable is Presburger arithmetic: https://en.wikipedia.org/wiki/Presburger_arithmetic https://en.wikipedia.org/wiki/Presburger_arithmetic Every question you have about its sentences will be answered by the theory one way or the other by either proving the statement or its negation.