Gentzen’s consistency proof for arithmetic
E761263
Gentzen’s consistency proof for arithmetic is a landmark 1930s result in proof theory that established the consistency of Peano arithmetic using transfinite induction up to the ordinal ε₀.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Die Widerspruchsfreiheit der reinen Zahlentheorie | 1 |
| Gentzen’s consistency proof for arithmetic canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8850216 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gentzen’s consistency proof for arithmetic Context triple: [Hilbert’s second problem, connectedToResult, Gentzen’s consistency proof for arithmetic]
-
A.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
-
B.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
-
C.
Lectures on the Logic of Arithmetic
Lectures on the Logic of Arithmetic is an educational work by Mary Everest Boole that explores the foundations and teaching of arithmetic through logical and psychological principles.
-
D.
Hilbert-style deductive systems
Hilbert-style deductive systems are axiomatic proof systems in mathematical logic that use a small set of axiom schemas and a few inference rules (typically including modus ponens) to derive theorems in formal theories such as Zermelo–Fraenkel set theory.
-
E.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gentzen’s consistency proof for arithmetic Target entity description: Gentzen’s consistency proof for arithmetic is a landmark 1930s result in proof theory that established the consistency of Peano arithmetic using transfinite induction up to the ordinal ε₀.
-
A.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
-
B.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
-
C.
Lectures on the Logic of Arithmetic
Lectures on the Logic of Arithmetic is an educational work by Mary Everest Boole that explores the foundations and teaching of arithmetic through logical and psychological principles.
-
D.
Hilbert-style deductive systems
Hilbert-style deductive systems are axiomatic proof systems in mathematical logic that use a small set of axiom schemas and a few inference rules (typically including modus ponens) to derive theorems in formal theories such as Zermelo–Fraenkel set theory.
-
E.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
consistency proof
ⓘ
mathematical proof ⓘ result in proof theory ⓘ |
| analyzes | structure of arithmetic proofs ⓘ |
| appliesTo | Peano arithmetic NERFINISHED ⓘ |
| assumes | primitive recursive arithmetic is sound ⓘ |
| author | Gerhard Gentzen NERFINISHED ⓘ |
| basedOn | sequent calculus ⓘ |
| concerns | first-order Peano arithmetic (PA) ⓘ |
| demonstrates | consistency of PA cannot be proved by purely finitist means alone (given Gentzen’s methods) ⓘ |
| establishes | termination of reduction process for proofs in arithmetic ⓘ |
| field |
foundations of mathematics
ⓘ
mathematical logic ⓘ proof theory ⓘ |
| goal | to prove the consistency of Peano arithmetic ⓘ |
| goesBeyond | Hilbert’s finitist program NERFINISHED ⓘ |
| historicalPeriod | 1930s ⓘ |
| impact | established proof theory as a central area of logic ⓘ |
| influenced |
ordinal analysis
ⓘ
proof-theoretic ordinal research ⓘ subsequent consistency proofs for stronger theories ⓘ |
| introduces | cut-elimination method ⓘ |
| isNonFinitist | true ⓘ |
| keyIdea | assign ordinals < ε₀ to proofs and show reduction decreases them ⓘ |
| languageOfOriginalPublication | German ⓘ |
| methodType | proof-theoretic consistency proof ⓘ |
| predecessorOf | later ordinal analyses of stronger systems ⓘ |
| preSupposes | consistency of transfinite induction up to ε₀ ⓘ |
| publishedIn | Mathematische Annalen NERFINISHED ⓘ |
| relatedTo |
Gödel’s incompleteness theorems
NERFINISHED
ⓘ
Hilbert’s program NERFINISHED ⓘ |
| relativeTo | transfinite induction up to ε₀ ⓘ |
| reliesOn | cut-elimination theorem NERFINISHED ⓘ |
| requires | well-foundedness of ordinals below ε₀ ⓘ |
| shows | no derivation of contradiction in Peano arithmetic ⓘ |
| status | classical result in proof theory ⓘ |
| subjectOf | consistency of Peano arithmetic ⓘ |
| titleOfPublication | Die Widerspruchsfreiheit der reinen Zahlentheorie NERFINISHED ⓘ |
| usesConcept |
induction along well-orders
ⓘ
measure of proof complexity by ordinals ⓘ ordinal notation system up to ε₀ ⓘ primitive recursive ordinal notation ⓘ proof-theoretic reduction ⓘ |
| usesFormalism |
first-order arithmetic
ⓘ
sequent calculus LK ⓘ |
| usesMethod | transfinite induction ⓘ |
| usesOrdinal | epsilon_0 (ε₀) ⓘ |
| yearProposed | 1936 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gentzen’s consistency proof for arithmetic Description of subject: Gentzen’s consistency proof for arithmetic is a landmark 1930s result in proof theory that established the consistency of Peano arithmetic using transfinite induction up to the ordinal ε₀.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Die Widerspruchsfreiheit der reinen Zahlentheorie