Linked from
The 58 pages that link to Gödel's incompleteness theorems, each with the reason it gives.
David HilbertCompared with: They sharply limited the consistency goals Hilbert had set for formal mathematics.
First-order logicRelated: They limit what particular first-order theories of arithmetic can prove, despite logic's completeness theorem.
John von NeumannCompared with: They revealed limits in the formalist ambitions that shaped von Neumann’s early mathematical work.
Set theoryNarrower topic: They place foundational theories, including set theory, within broader limits on formal proof.
Bertrand RussellRelated: Gödel’s results limited the ambitions of the kind of formal foundations Russell pursued.
Classical logicRelated: They limit what formal mathematical systems based on classical reasoning can establish.
Hilbert's programRelated: They ruled out key ambitions of Hilbert's program for arithmetic-capable systems.
Formal proofRelated: They show that formal derivability has limits even in powerful mathematical systems.
Principia MathematicaRelated: They constrained hopes that a formal system like Principia Mathematica could settle all foundational questions.
Proof theoryCompared with: They sharply constrained the consistency ambitions that motivated early proof theory.
ConsistencyRelated: They sharply constrain what such systems can establish about their own consistency and completeness.
Gödel numberingBroader topic: Gödel used number codes to express statements about formal proofs within arithmetic.
Mathematical logicRelated: They expose limits on what formal mathematical theories can establish from within.
Kurt GödelBroader topic: These are the central results Gödel established using arithmetized syntax.
SoundnessCompared with: They concern limits of provability, not failures of soundness.
Formal systemRelated: They establish that some formal systems cannot prove every relevant truth about arithmetic.
Peano arithmeticRelated: Peano arithmetic is subject to incompleteness because it can represent enough elementary computation.
EntscheidungsproblemCompared with: They concern limits of formal proof, while the Entscheidungsproblem concerns algorithmic decision of validity.
Computable functionRelated: Their proofs use effective formalization and connect computation with limits of proof.
Formal logicRelated: They reveal limits on completeness and self-verification in formal systems.
Foundations of mathematicsRelated: They sharply limited the ambitions of Hilbert's program.
TheoremBroader topic: They establish limits on which mathematical statements formal systems can prove.
Axiomatic methodRelated: They established that axiomatization cannot provide every desired guarantee for arithmetic.
DiagonalizationRelated: Gödel's construction uses arithmetized self-reference to create an unprovable sentence.
FormalismRelated: They showed that Hilbert's consistency ambitions could not be achieved by the intended finitary means.
Diagonal argumentRelated: Gödel uses formal self-reference, built with diagonal reasoning, to produce an unprovable sentence.
Gödel's completeness theoremCompared with: Despite the shared name, incompleteness concerns particular theories, not first-order logic's proof calculus.
LogicismRelated: They constrain hopes that one fixed formal system can capture all mathematical truth.
CompletenessCompared with: They concern truth in intended arithmetic, not completeness for all models.
Diagonal lemmaRelated: The first incompleteness theorem uses the diagonal lemma to build an undecidable sentence.
Independence (mathematical logic)Related: They explain why consistent, sufficiently strong systems cannot decide every sentence in their language.
Presburger arithmeticRelated: Presburger arithmetic avoids their force by lacking the expressive power of full arithmetic.
Saul KripkeRelated: Kripke’s Gödel cases test whether names refer through descriptions or through causal-historical links.
Self-referenceBroader topic: Gödel encoded a sentence that, informally, says it is not provable in its system.
Foundations of GeometryNarrower topic: They constrain what foundational systems can prove about their own consistency and completeness.
Hilbert's tenth problemRelated: They helped transform expectations about formal methods during the decades after Hilbert posed his problems.
Peano axiomsRelated: They limit what first-order Peano arithmetic can settle about arithmetic.
Philosophy of mathematicsRelated: They limit what formal foundations can prove about arithmetic and themselves.
Axiomatic systemRelated: They show that sufficiently expressive systems cannot prove every arithmetic truth or establish their own consistency in the required setting.
Deductive systemRelated: They limit what consistent deductive systems for arithmetic can prove.
History of logicBroader topic: They changed expectations about whether mathematics could be captured by a complete formal system.
Goodstein's theoremNarrower topic: Goodstein's theorem is a concrete arithmetic instance of the limits these results predict.
Arithmetical hierarchyRelated: Arithmetical definability helps describe the complexity of truths and unprovable statements.
Gödel sentenceNarrower topic: The first theorem uses a Gödel sentence to establish incompleteness.
Hilbert's problemsRelated: They transformed expectations surrounding foundational questions associated with Hilbert's program.
Large cardinalRelated: They help explain why existence claims stronger than a theory's axioms may remain undecidable within it.
Tarski's undefinability theoremRelated: Both results use arithmetized syntax and self-reference, though they establish different limitations.
Kleene's recursion theoremCompared with: Their self-reference is used to establish unprovability, not to construct behaviorally fixed-point programs.
Paris–Harrington theoremRelated: The theorem gives a concrete combinatorial instance of incompleteness.
Formal scienceBroader topic: They demonstrate fundamental limits on what formal proof systems can establish.