Axiomatic system

An axiomatic system consists of axioms and rules of inference used to derive theorems. Its formal language and derivation rules determine which statements can be proved within it.

Axiom: A statement accepted as a starting assumption within a deductive system. Axioms supply the premises from which the system's theorems are derived.
Deductive closure: The set of all statements derivable from a given set of premises under specified inference rules. It describes the full collection of results generated by the system's axioms.
Gödel's incompleteness theorems: Two theorems showing limits on provability in consistent formal systems capable of expressing arithmetic. They show that sufficiently expressive systems cannot prove every arithmetic truth or establish their own consistency in the required setting.
Natural deduction: A proof system organized around introduction and elimination rules for logical connectives and quantifiers. It presents inference through rules for reasoning with assumptions rather than a fixed axiom list.
Rule of inference: A formal rule specifying how conclusions may be derived from premises. Inference rules determine which steps count as valid derivations in the system.
Soundness: A property of a proof system in which every provable statement is valid under its semantics. Soundness ensures that derivations do not establish semantically invalid conclusions.
Undecidable proposition: A statement that a formal system can neither prove nor refute, assuming the system is consistent. Such propositions expose gaps between what is true in an intended model and what the system derives.
Sequent calculus: A formal proof system whose judgments are sequents relating collections of assumptions to conclusions. It offers a different proof architecture for formalizing deduction.
Formal language: A language with precisely specified symbols and rules for forming expressions. Its syntax sets the expressions that can serve as axioms or theorems.
Completeness: A property in which every statement valid under a specified semantics is provable in a proof system. Completeness measures whether the system can derive every semantic consequence it should capture.

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