Knowra Gödel sentence Gödel sentence A sentence constructed within a formal system to assert, through arithmetic coding, that it is not provable in that system. Under suitable conditions, it is true but unprovable there.
Gödel numbering : A method that assigns natural numbers to symbols, formulas, and proofs in a formal language. It lets a sentence describe its own proof status using arithmetic.
First-order logic : A formal logic with variables, predicates, quantifiers, and rules for reasoning about objects. Gödel sentences are formulated inside theories expressed in a formal language such as this.
Gödel's incompleteness theorems : Two theorems showing that sufficiently expressive, effectively axiomatized consistent theories cannot prove every arithmetic truth or their own consistency. The first theorem uses a Gödel sentence to establish incompleteness.
Gödel's 1931 proof : Kurt Gödel’s proof that sufficiently expressive, consistent, effectively axiomatized formal systems contain undecidable arithmetic sentences. Its construction gives the original setting and argument behind the Gödel sentence.
Diagonal lemma : A result guaranteeing a sentence that is equivalent, within a theory, to a formula applied to its own code. It supplies the formal self-reference used to construct a Gödel sentence.
Peano arithmetic : An axiomatic theory describing natural numbers through zero, successor, addition, and multiplication. It is a standard setting rich enough to encode its own formulas and proofs.
Completeness (logic) : The property that every sentence true in all models of a theory is provable from that theory. Logical completeness does not prevent a particular arithmetic theory from being incomplete.
Rosser's theorem : A refinement of Gödel’s first incompleteness theorem that requires consistency rather than the stronger ω-consistency assumption. Rosser altered the self-referential sentence to obtain a sharper independence result.
Provability predicate : A formula in arithmetic that represents whether a sentence has a formal proof in a specified theory. The sentence asserts its unprovability by negating this represented proof relation.
Recursive axiomatizability : The property that a formal theory’s axioms can be enumerated by an effective procedure. Effective axiom enumeration supports the arithmetical representation of provability.
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