Consistency

A formal theory is consistent if no statement and its negation are both derivable from its axioms. In classical logic, this is equivalent to deriving no contradiction.

Formal system: A formal system consists of symbols, formation rules, axioms, and rules of inference. Consistency is defined relative to a system’s axioms and inference rules.
Formal language: A formal language is a precisely specified collection of symbols and rules for forming expressions. Consistency applies to formulas built in a language with a defined negation.
David Hilbert: David Hilbert was a German mathematician who shaped modern axiomatic mathematics and proposed a program for securing its foundations. His program made proofs of consistency a central foundational goal.
Gödel's incompleteness theorems: Gödel's incompleteness theorems show that sufficiently expressive, effectively axiomatized consistent systems cannot settle every arithmetical statement. They sharply constrain what such systems can establish about their own consistency and completeness.
Completeness (logic): A theory is syntactically complete if it proves every sentence in its language or its negation. A theory can be consistent yet leave some sentences undecided.
Derivation: A derivation is a finite sequence of formulas, each justified by an axiom or an inference rule. A contradiction threatens consistency only when both it and its negation have derivations.
Axiom: An axiom is a sentence accepted as a starting assumption in a formal theory. Changing a theory’s axioms can change whether it has a contradiction.
Hilbert's program: Hilbert's program sought to justify mathematics using finitary methods to prove the consistency of formalized theories. It treated consistency proofs as a route to confidence in formal mathematics.
Gödel's second incompleteness theorem: Gödel's second incompleteness theorem states that a suitable consistent formal theory cannot prove its own formalized consistency. It limits internal consistency proofs for theories capable of expressing arithmetic.
Soundness (logic): A proof system is sound when every sentence it derives is valid under its intended semantics. Soundness concerns correctness of derivations, while consistency concerns avoiding contradictory derivations.

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