Note: see Gödel second theorem.

Important Properties of Formal Systems

Consistency

A system is consistent if it is impossible, within the system, to derive both a statement and its negation. A system is inconsistent if a statement and its negation are both derivable.

In an inconsistent system, every possible statement (of its language) can be derived as a theorem, because everything can be logically derived from a contradiction. In a consistent system, that isn't the case.

Consistency is also referred to as the correctness or soundness of the system.

Completeness

A complete system will have a derivable theorem to correspond to every true formula in its language. An incomplete system will not be able to derive some true formula.

Decidability

A system is decidable if every true formula of the system has a proof and every untrue formula of the system has a disproof. A system is undecidable if there is some statement that it can neither prove nor disprove.

Note: The properties described here are stated in terms of logical systems; they can also be stated for formal systems more generally. For discussion of formal systems and their properties see Smullyan (1961). For further discussion of the properties of logical systems, see Hunter (1973), Anderton (1972), or Smulyan (1961).