The word "decidable" has two distinct meanings in mathematics. The first meaning came from logic. A set of axioms is a set of statements that are regarded as true in the system for which the axioms hold. If a statement in the language of the system can be proven true or false within the system itself, then it is said to be decidable. For example, the statement "1 + 1 = 2" can be proven true in Zermelo-Fraenkel set theory. So, this statement is decidable in Zermelo-Fraenkel set theory. On the other hand, the continuum hypothesis is undecidable in Zermelo-Fraenkel set theory. Gödel's incompleteness theorem states that for most systems (those whose axioms are defined recursively), the consistency of the system is undecidable within itself. This means that it cannot be proven, by using only the axioms within the system, that internal contradictions do not occur, i.e. that no statement...