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 |
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 ⓘ |
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