Gödel's incompleteness theorem

URI: https://gptkb.org/entity/Gödel's_incompleteness_theorem
GPTKB entity
Statements (23)
Predicate Object
gptkbp:instanceOf gptkb:mathematical_concept
gptkbp:appliesTo gptkb:Zermelo–Fraenkel_set_theory
gptkb:Peano_arithmetic
gptkbp:field gptkb:logic
gptkb:mathematics
gptkbp:firstTheorem In any consistent formal system adequate for arithmetic, there are statements that are true but unprovable within the system.
gptkbp:formedBy gptkb:Kurt_Gödel
https://www.w3.org/2000/01/rdf-schema#label Gödel's incompleteness theorem
gptkbp:influenced gptkb:logic
gptkbp:language gptkb:German
gptkbp:numberOfTheorems 2
gptkbp:originalTitle gptkb:Über_formal_unentscheidbare_Sätze_der_Principia_Mathematica_und_verwandter_Systeme_I
gptkbp:publicationYear 1931
gptkbp:publishedIn gptkb:Monatshefte_für_Mathematik_und_Physik
gptkbp:relatedTo gptkb:Church–Turing_thesis
gptkb:Hilbert's_program
gptkb:Turing's_halting_problem
gptkbp:secondTheorem No consistent system can prove its own consistency.
gptkbp:sentence Any consistent formal system that is sufficiently expressive cannot be both complete and consistent.
There exist true statements in arithmetic that cannot be proven within the system.
gptkbp:bfsParent gptkb:Über_formal_unentscheidbare_Sätze_der_Principia_Mathematica_und_verwandter_Systeme_I
gptkb:The_Emperor's_New_Mind
gptkbp:bfsLayer 5